Ratios, Rates, and Percents

Unit 4 - Mrs. Jenkins Math

Ratios, Rates, and Percents Vocabulary

Here are our Unit 4 Vocabulary Words

Ratios

Textbook Pages 149-152


Learning Target:
I understand and can write ratios to make comparisons in real-world contexts.


Essential Questions: What are ratios? How can we use ratios to make real-world comparisons?

Read about Ratios

(From PurpleMath.com)


Proportions are built from ratios. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20.


Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, its number must come first. If the expression had been "the ratio of women to men", then the numbers would have been "20 to15".


Expressing the ratio of men to women as "15 to20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:

    odds notation: 15 : 20

    fractional notation: 15/20


You should be able to recognize all three notations; you will probably be expected to know them for your test.


Given a pair of numbers, you should be able to write down the ratios. For example:

  • There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese in all three formats.
  • 16:9, 16/9, and 16 to 9
  • Consider the above park. Express the ratio of geese to ducks in all three formats.
  • 9:16, 9/16, and 9 to 16


The numbers were the same in each of the above exercises, but the order in which they were listed differed, varying according to the order in which the elements of the ratio were expressed. In ratios, order is very important.


Let's return to the 15 men and 20 women in our original group. I had expressed the ratio as a fraction, namely, 15/20. This fraction simplifies to 3/4. This means that you can also express the ratio of men to women as 3/4, 3 : 4, or "3 to 4".


This points out something important about ratios: the numbers used in the ratio might not be the absolute measured values. The ratio "15 to 20" refers to the absolute numbers of men and women, respectively, in the group of thirty-five people. The simplified ratio "3 to 4" tells you only that, for every three men, there are four women. The simplified ratio also tells you that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men. In other words, the men comprise 3/7 of the people in the group. These relationships and reasoning are what you use to solve many word problems:

  • In a certain class, the ratio of passing grades to failing grades is 7 to 5. How many of the 36 students failed the course?

    The ratio, "7 to 5" (or 7 : 5 or 7/5), tells me that, of every 7 + 5 = 12 students, five failed. That is, 5/12 of the class flunked. Then ( 5/12 )(36) = 15 students failed.

  • In the park mentioned above, the ratio of ducks to geese is 16 to 9. How many of the 300 birds are geese?

    The ratio tells me that, of every 16 + 9 = 25 birds, 9 are geese. That is, 9/25 of the birds are geese. Then there are ( 9/25 )(300) = 108 geese.


Generally, ratio problems will just be a matter of stating ratios or simplifying them. For instance:

  • Express the ratio in simplest form: $10 to $45

    This exercise wants me to write the ratio as a simplified fraction:

      .10/45 = 2/9.


This simplified fraction is the ratio's expression in simplest fractional form. Note that the units (the "dollar" signs) "canceled" on the fraction, since the units, "$", were the same on both values. When both values in a ratio have the same unit, there should generally be no unit on the reduced form.

  • Express the ratio in simplest form: 240 miles to 8 gallons

    When I simplify, I get (240 miles) / (8 gallons) = (30 miles) / (1 gallon), or, in more common language, 30 miles per gallon.


In contrast to the answer to the previous exercise, this exercise's answer did need to have units on it, since the units on the two parts of the ratio, the "miles" and the "gallons", do not "cancel" with each other.


Simplified ratios are equivalent ratios. Equivalent ratios are ratios that name the same comparison. You can find equivalent ratios by either multiplying or dividing both terms in the ratio by the same number.


Conversion factors are simplified ratios, so they might be covered around the same time that you're studying ratios and proportions. For instance, suppose you are asked how many feet long an American football field is. You know that its length is 100 yards. You would then use the relationship of 3 feet to 1yard, and multiply by 3 to get 300 feet.


Ratios are the comparison of one thing to another (miles to gallons, feet to yards, ducks to geese, et cetera). But their true usefulness comes in the setting up and solving of proportions....

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Equal Ratios | All Those Different Size Screens | PBSMathClub
Introduction to ratios | Ratios, proportions, units, and rates | Pre-Algebra | Khan Academy
Ratio word problem: boys to girls | Pre-Algebra | Khan Academy
Proportions | Baking Beet Cookies? | PBSMathClub
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Rates

Textbook Pages 155-158


Learning Target: I understand and can apply rates in real world situations.


Essential Questions: What are rates? How do we use rates in real world situations?

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Read about Rates

A ratio is a comparison of two numbers or measurements. The numbers or measurements being compared are called the terms of the ratio. A rate is a special ratio in which the two terms are in different units. For example, if a 12-ounce can of corn costs 69¢, the rate is 69¢ for 12 ounces. The first term of the ratio is measured in cents; the second term in ounces.


You can write this rate as 69¢/12 ounces or 69¢:12 ounces. Both expressions mean that you pay 69¢ "for every" 12 ounces of corn.


Rates are used by people every day, such as when they work 40 hours a week or earn interest every year at a bank. When rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates. If you have a multiple-unit rate such as 120 students for every 3 buses, and want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term.


120/3 = 40/1


The unit rate of 120 students for every 3 buses is 40 students per bus. You could also find the unit rate by dividing the first term of the ratio by the second term. When prices are expressed as a quantity of 1, such as $25 per ticket or $0.89 per can, they are called unit prices. If you have a multiple-unit price, such as $5.50 for 5 pounds of potatoes, and want to find the single-unit price, divide the multiple-unit price by the number of units.


$5.50 ÷ 5 = $1.10


The unit price of potatoes that cost $5.50 for 5 pounds is $1.10 per pound.


Rates and unit rates are used to solve many real-world problems. Look at the following problem.


  • "Tonya works 60 hours every 3 weeks. At that rate, how many hours will she work in 12 weeks?" The problem tells you that Tonya works at the rate of 60 hours every 3 weeks.
  • To find the number of hours she will work in 12 weeks, write a ratio equal to 60/3 that has a second term of 12.
  • 60/3 = 240/12
  • Tonya will work 240 hours in 12 weeks.


You could also solve this problem by first finding the unit rate and multiplying it by 12.


  • 60/3 = 20/1
  • 20 x 12 = 240


When you find equal ratios, it is important to remember that if you multiply or divide one term of a ratio by a number, then you need to multiply or divide the other term by that same number. Now let's take a look at a problem that involves unit price.


  • "A sign in a store says 3 Pens for $2.70. How much would 10 pens cost?"
  • To solve the problem, find the unit price of the pens, then multiply by 10.
  • $2.70 ÷ 3 = $0.90
  • $0.90 10 = $9.00


Finding the cost of one unit first makes it easy to find the cost of multiple units.

Ratios and Rates - Konst Math
Solving unit rates problem | Ratios, proportions, units, and rates | Pre-Algebra | Khan Academy
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Ratio and Rate Reasoning

Textbook Pages 161-164


Learning Target: I can use ratio and rate reasoning to solve problems by creating tables, double number lines, graphs, and proportions.


Essential Question: How can tables, double number lines, graphs, and proportions help me use ratio/rate reasoning to solve problems?

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Read about Equivalent Ratios and Rates

A rate is a ratio that compares quantities in different units. The word "per" is used when talking about rates and is sometimes abbreviated with a slash, /.


Some common rates are used all the time, like miles per hour, dollars per gallon, and days per week. Others are more specific to an occasion, like cats per household or chocolate chips per cookie.


Two rates are equivalent if they show the same relationship between two identical units of measure.


The same strategies used to find equivalent ratios can be used to find equivalent rates.

Here is an example:


  • Two cars are traveling. One car goes 40 miles in 2 hours, and the other goes 80 miles in 4 hours. Determine whether or not the rates are equivalent.
  • First, write the rates as fractions. Remember to make sure the terms are the same, in this case, miles per hour.
  • 40 miles per 2 hours = 40 miles / 2 hours = 40/2
  • 80 miles per 4 hours = 80 miles / 4 hours = 80/4
  • Next, simplify both fractions to their lowest terms.
  • 40/2 = 40÷2 / 2÷2 = 20 miles / 1 hour
  • 80/4 = 80÷4 / 4÷4 = 20 miles / 1 hour
  • Then, compare the fractions in their lowest terms. Remember to include the units.
  • 20 miles / 1 hour = 20 miles / 1 hour
  • The answer is that the rates are equivalent. Both cars traveled at a rate of 20 miles per hour.


Other methods that can be used to determine whether or not rates are equal include changing one or both fractions so that the denominators are equal, and cross multiplying.

Cross multiplication does not yield a rate. Only the numerical values are used as a check.

When using rates, it does not matter which unit is in the numerator and which is in the denominator, as long as they match one another.

For example:


  • gumdrops per person is not the same as persons per gumdrop
  • gumdrops / person ≠ persons / gumdrop


Guided Practice

Determine if two machines are winding thread at the same speed. One machine winds at a rate of 5 meters every 3 seconds. The other takes 18 seconds to wind 20 meters.

  • First, write the rates as fractions, making sure the units are identical.
  • 53 meters / 1 second and 20 meters / 18 seconds
  • Next, assume the rates are equal to one another and write an equation.
  • 5/3 = 20/18
  • Then cross multiply.
  • 5 x 18 = 3 x 20
  • 90≠60
  • The answer is that the rates are not equivalent.


Example 1

Are the rates equivalent? 3 feet in 9 seconds and 6 feet in 18 seconds

  • First, write the rates as fractions.
  • 3/9 and 6/18
  • Next, determine a method. Multiply the first fraction times 2 to convert to, and work with, 18ths.
  • 3/9 = 3×2 / 9×2 = 6/18
  • Then, compare the fractions.
  • 6 feet / 18 seconds = 6 feet / 18 seconds
  • The answer is that the rates are equivalent.


Example 2

Compare the rates: 5 miles in 30 minutes and 42 minutes to go 6 miles

  • First, write the rates as fractions being sure to keep the units consistent.
  • 5 miles / 30 minutes and 6 miles / 42 minutes
  • Next, determine a method. Simplify both rates to their lowest terms.
  • 5/30 = 5÷5 / 30÷5 = 1 mile / 6 minutes
  • 6/42 = 6÷6 / 42÷6 = 1 mile / 7minutes
  • Then, compare the fractions.
  • 1/6 ≠ 1/7
  • The answer is that the rates are not equal.


Example 3

Compare 5 pounds for $20.00 and 8 pounds for $32.00

  • First, write the rates as fractions.
  • 5 pounds / 20 dollars and 8 pounds / 32 dollars
  • Next, determine a method. Simplify each fraction to its lowest terms.
  • 5/20 = 5÷5 / 20÷5 = 1 pound / 4 dollars
  • 8/32 = 8÷8 / 32÷8 = 1 pound / 4 dollars
  • Then, compare the rates.
  • 1/4 = 1/4
  • The answer is that the rates are equivalent.
  • 1 pound / 4 dollars can also be written 4 dollars / 1 pound or $4 per pound.
Comparing Ratios 4 Methods With Word Problems
Comparing and Graphing Ratios
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Read About Scale

A scale is a ratio that shows the relationship between the representation of an object and the real measurement of an object. Toy makers use scale ratios to make models of cars and airplanes that are proportional to the real thing. Architects use scale drawings to plan the design of buildings.


A scale drawing is a drawing that is used to represent and object that is too large to be drawn in its actual dimensions. If a drawing is to scale, you can use proportions to determine the actual dimensions of the scale drawing.


Let’s look at a scale drawing of a sculpture. In the drawing, the sculpture is 4 inches tall. The scale is 1 inch : 4 feet. Its scale can be written as 1 in. / 4 ft. A drawing with this scale tells you that 1 inch on paper is equal to 4 feet in real life. If the drawing is 4 inches tall, use the scale to find the actual height of the sculpture.

  • First, create a proportion using equivalent ratios. Remember to write the corresponding units in the numerator and in the denominator. The actual height of the sculpture is represented by the variable x.
  • 1 in. / 4 ft. = 4 in. / x ft.
  • Then, cross multiply and simplify the equation to find the value of x.
  • 1 × x = 4 × 4
  • x = 16
  • The sculpture will be 16 feet tall.


Let’s make a scale drawing. Use the scale 1 in. = 2 ft. Draw a room that is 8 feet × 12 feet.

  • First, write a proportion to find the measurement of the width.
  • 1 in. / 2 ft. = x in. / 8 ft.
  • Next, cross multiply and simplify the equation to find the value of x.
  • 1 × 8 = 2 × x
  • 8 = 2 × x
  • x = 4
  • The drawing will be 4 inches wide.
  • Then, write a proportion to find the measurement of the length.
  • 1 in. / 2 ft. = x in. / 12 ft.
  • Finally, cross multiply and simply the equation to find the value of x.
  • 1 × 12 = 2 × x
  • 12 = 2 × x
  • x = 6
  • The drawing will be 6 inches long.
  • In the drawing room will be 4 inches × 6 inches.


Scale dimensions are also used to figure out the actual dimensions of something. The flower bed design shows that the width of the garden on the drawing is six inches. If the scale is 1 in. = 5 ft, how wide is the actual flower garden?

  • First, write a proportion to find the actual measurement of the flower bed.
  • 1 in. / 5 ft. = 6 in. / x ft.
  • Then, cross multiply and simplify the equation to find the value of x.
  • 1 × x = 5 × 6
  • x = 30
  • The actual flower bed is 30 feet wide.


Guided Practice

Solve the proportion.

  • 7 in. / 70 ft. = x in. / 140 ft.
  • First, cross multiply and simplify to find the value of x.
  • 7 × 140 = 70 × x
  • 980 = 70 × x
  • Divide both sides of the equation by 70.
  • x × 70 ÷ 70 = x
  • 980 ÷ 70 = 14
  • x = 14
  • Or, use mental math to solve for x.
  • First, look at the relationship between the two given denominators. Think, “The second denominator (140 ft.) is double (2 ×) the first denominator (70 ft.).”
  • Then, take the given numerator and multiply by 2. Think, “7 times 2 is 14.”
  • The answer is 14 inches.


Examples

Solve each of the following proportions.


Example 1: 1 in. / 3 ft. = x in. / 21 ft.

  • First, cross multiply and simplify to find the value of x.
  • 1 × 21 = 3 × x
  • 21 = 3 × x
  • x = 7
  • The answer is 7 inches.


Example 2: 3 in. / 6 ft. = 9 in. / x ft.

  • First, cross multiply and simplify to find the value of x.
  • 3 × x = 6 × 9
  • 3 × x = 54
  • Divide both sides of the equation by 3.
  • x × 3 ÷ 3 = x
  • 54 ÷ 3 = 18
  • x = 18
  • The answer is 18 feet.


Example 3: 2 in. / 10 ft. = x in. / 120 ft.

  • First, cross multiply and simplify to find the value of x.
  • 2 × 120 = 10 × x
  • 240 = 10 × x
  • Divide both sides by 10.
  • x × 10 ÷ 10 = x
  • 240 ÷ 10 = 24
  • x = 24
  • The answer is 24 inches.


Now that you have figured out how to use proportions to figure out actual and scale dimensions, we can look at figuring out scale factors.


What is a scale factor?

  • A scale factor is another name for a scale ratio. When looking for a scale factor, you can look at the relationship between the scale measurement and the actual measurement to determine what scale was used. This scale is called the scale factor.


A fence is actually 16 feet long. If the fence is drawn as four inches, what is the scale factor?

  • To figure this out, we need to write a ratio to compare the drawing of the fence to the actual measurement.
  • 4 in. / 16 ft.
  • Now we want to figure out the scale factor. To do this, we simplify the ratio using the greatest common factor. The greatest common factor of 4 and 16 is 4.
  • 4÷4 / 16÷4 = 1 / 4
  • The scale factor is 1 in. / 4 ft.

Use this information to simplify and find the following scale factors.

Example A: 5 in. / 25 ft.

  • Solution: 5÷5 / 25÷5 = 1 in. / 5 ft.

Example B: 2 in. / 50 ft.

  • Solution: 2÷2 / 50÷2 = 1 in. / 25 ft.

Example C: 12.5 in. / 25 ft.

  • Solution: 12.5÷12.5 / 25÷12.5 = 1 in. / 2 ft.


At the supermarket, Sarah was looking at the cereal boxes as she was putting them away. On the back of one of the boxes, there was a picture of a skyscraper. Underneath the picture was a ratio. It said: 1 inch / 30 feet

  • Sarah wasn't sure what this meant. Do you know?
  • Sarah has seen a scale ratio. Inches are being compared to feet. In other words, the scale is saying that for every inch of the building on the cereal box that it represents 30 feet of actual height.
  • What about scale factor?
  • We can figure out the scale factor if we can simplify this ratio. However, this ratio is already in simplest form so this scale ratio is also the scale factor.


Guided Practice

What is the scale factor: 3 / 12

  • Answer: To figure this out, we divide the denominator and the numerator by the same number (GCF). Once the fraction is simplified, it will show the scale factor: 1 / 4
Scale Model Problems
Scale Factor
Solve a scale drawing word problem | Geometry | 7th grade | Khan Academy
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Unit Conversions

Learning Target: I can convert within and between the metric and customary unit systems.


Essential Question: How can proportions help me convert within and between the metric and customary unit systems?

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Read About Converting Within the Customary System

When converting customary units of measure from a larger unit to a smaller unit, create a proportion using the unit rate of the conversion factor to find an equivalent rate. Here is an example to demonstrate this.


  • 1 dollar = 100 pennies or 100 pennies / 1 dollar


There are 100 pennies in one dollar. The dollar is a larger unit than the penny. You need many pennies to equal one dollar. The same is true when working with units of length, weight and capacity. You need more of a smaller unit to equal a larger unit.


Think back to all of the units of length, weight and capacity that you have previously learned about.


Let's look at a conversion problem.


John has a rope that is 10 feet long. How long is his rope in inches?

  • Notice, you are converting from feet to inches. A foot is a larger unit than an inch.
  • To solve this problem, create a proportion with the unit rate of how many inches in 1 foot and an equivalent rate for 10 feet. This will give you the measurement in inches.
  • 1 foot = 12 inches or 12 inches / 1 foot
  • 12 in. / 1 ft. = x in. / 10 ft.
  • The number of feet was multiplied by 10 to make 10 ft.; so, the number of inches must in 1 foot must be multiplied by 10 to get an equivalent rate for 10 feet.
  • 12 in. × 10 / 1 ft. × 10 = 120 in. / 10 ft.
  • The answer is 10 feet is equivalent to 120 inches.


Guided Practice

Convert the units of measure.


Jason’s baby brother drank 3 cups of milk. How many fluid ounces did he drink?

  • Once again, you are going from a larger unit to a smaller unit.
  • To solve this problem, create a proportion with the unit rate of how many ounces in 1 cup and an equivalent rate for 3 cups. This will give you the measurement in ounces.
  • 1 cup = 8 fl oz. or 8 fl oz. / 1 cup
  • 8 fl oz. / 1 cup = x fl oz. / 3 cups
  • To change 1 cup to 3 cups we must multiply by three. The same must happen to 8 fl oz.
  • 8 fl oz. × 3 / 1 cup × 3 = 24 fl oz. / 3 cups
  • The answer is 3 cups is equivalent to 24 fluid ounces.


Examples

Convert the following units of measure.


Example 1: 4 tons = ____ pounds

  • First create a proportion with the unit rate of how many pounds that are in 1 ton and find an equivalent rate for 4 tons. This will give you the measurement in pounds.
  • 1 ton = 2,000 pounds or 2,000 lb. / 1 T
  • 2,000 lb. / 1 T = x lb. / 4 T
  • The number of tons was multiplied by 4 to get 4 tons total. So, the number of pounds must be multiplied by 4 also.
  • 2,000 lb. × 4 / 1 T × 4 = 8,000 lb. / 4 T
  • The answer is 4 tons is equivalent to 8,000 pounds


Example 2: 5 feet = ____ inches

  • First create a proportion with the unit rate of how many inches that are in 1 foot and find an equivalent rate for 5 feet. This will give you the measurement in inches.
  • 1 foot = 12 inches or 12 in. / 1 ft.
  • 12 in. / 1 ft. = x in. / 5 ft.
  • The 1 foot was multiplied by 5 to get 5 feet. The number of inches needs to also be multiplied by 5.
  • 12 in. × 5 / 1 ft. × 5 = 60 in. / 5 ft.
  • The answer is 5 feet is equivalent to 60 inches


Example 3: 8 pints = ____ cups

  • First create a proportion with the unit rate of how many cups that are in 1 pint and find an equivalent rate for 8 pints. This will give you the measurement in cups.
  • 1 pint = 2 cups or 2 cups / 1 pint
  • 2 cups / 1 pint = x cups / 8 pints
  • The number of pints in the unit rate was multiplied by 8 to get 8 pints. So, the number of cups needs to be multiplied by 8 also.
  • 2 cups × 8 / 1 pints × 8 = 16 cups / 8 pints
  • The answer is 8 pints is equivalent to 16 cups.

To convert from a smaller unit to a larger unit, create a proportion using the unit rate of the conversion factor to find an equivalent rate. Let’s consider pennies.

  • 5,000 pennies = ____ dollars


The penny is a smaller unit than the dollar. You need more of a smaller unit to equal a larger unit, create a proportion using the unit rate of the conversion factor to find an equivalent rate. You know that there are 100 pennies in one dollar.

  • 100 pennies = 1 dollar or 100 pennies / 1 dollar
  • Set up the proportion: 100 pennies / 1 dollar = 5,000 pennies / x dollars
  • To determine what 100 pennies was multiplied to get 5,000 pennies, divide 5,000 by 100 = 50. So, 100 pennies was multiplied by 50 to get 5,000 pennies. That means that the 1 dollar must also be multiplied by 50, which equals 50 dollars.
  • 100 pennies × 50 / 1 dollar × 50 = 5,000 pennies / 50 dollars
  • Therefore, 5,000 pennies is equivalent to 50 dollars.


Apply this to your work with converting measurements. Remember to think about the equivalent units of length, capacity and weight when dividing.


Let's look at a conversion problem: 5,500 pounds = ____ tons

  • A pound is smaller than a ton. To solve this problem, create a proportion using the unit rate of the conversion factor to find an equivalent rate.
  • 2,000 pounds = 1 ton or 2,000 lb. / 1 T
  • 2,000 lb. / 1 T = 5,500 lb. / x T
  • To figure out what 2,000 lb. was multiplied to get 5,500 lb. you must divide 5,500 by 2,000 = 2.75. This means that 2,000 lb. was multiplied by 2.75 to get 5,500 lb. We must then do the same to the 1 T.
  • 2,000 lb. × 2.75 / 1 T × 2.75 = 5,500 lb. / 2.75 T
  • Therefore, 5,500 pounds is equivalent to 2.75 tons.


Guided Practice

Convert the unit of measure.


1,250 inches = _____ feet

  • To solve this problem, convert inches to feet. Create a proportion using the unit rate of the conversion factor to find an equivalent rate.
  • 12 inches = 1 foot or 12 in. / 1 ft.
  • 12 in. / 1 ft. = 1,250 in. / x ft.
  • Divide 1,250 by 12 to determine what 12 was multiplied by to get 1,250. 1,250 ÷ 12 = 104.16.
  • 12 ft. × 104.16 / 1 ft. × 104.16 = 1,250 in. / 104.16 ft.
  • The answer is 1,250 inches is equivalent to 104.16 feet.


Examples

Convert the following units of measure.


Example 1: 84 inches = ____ feet

  • Create a proportion using the unit rate of the conversion factor to find an equivalent rate.
  • 12 inches = 1 foot or 12 in. / 1 ft.
  • 12 in. / 1 ft. = 84 in. / x ft.
  • 84 ÷ 12 = 7
  • 12 in. × 7 / 1 ft. × 7 = 84 in. / 7 ft.
  • The answer is 84 inches is equivalent to 7 feet.


Example 2: 40 cups = ____ pints

  • Create a proportion using the unit rate of the conversion factor to find an equivalent rate.
  • 2 cups = 1 pint or 2 cups / 1 pint
  • 2 cups / 1 pint = 40 cups / x pints
  • 40 ÷ 2 = 20
  • 2 cups × 20 / 1 pint × 20 = 40 cups / 20 pints
  • The answer is 40 cups is equivalent to 20 pints.


Example 3: 800 pounds = ____ tons

  • Create a proportion using the unit rate of the conversion factor to find an equivalent rate.
  • 2,000 pounds = 1 ton or 2,000 lb. / 1 T
  • 2,000 lb. / 1 T = 800 lb. / x T
  • 800 ÷ 2,000 = 0.4
  • 2,000 lb. × 0.4 / 1 T × 0.4 = 800 lb. / 0.4 T
  • The answer is 800 pounds is equivalent to 0.4 tons.

Read About Converting Within the Metric System

Converting within the Metric System is the same process as converting within the Customary System. You just need to write the conversion factors as unit rates; then, create proportions to find equivalent rates.


Convert the units of measurement: 12,350 mL= ____ L

  • First, identify the unit conversion. You are converting a smaller unit to a larger unit. You know that a 1,000 milliliters is one liter, or 1,000 mL / 1 L.
  • Create the proportion using the unit rate.
  • 1,000 mL / 1 L = 12,350 mL / x L
  • To determine what 1,000 was multiplied by to get 12,350 you must divide 12,350 by 1,000 which equals 12.35. So both 1,000 mL and 1 L were multiplied by 12.35 in the new equivalent rate.
  • 1,000 mL × 12.35 / 1 L × 12.35 = 12,350 mL / 12.35 L
  • 12,350 milliliters is equal to 12.35 liters.


Examples

Covert the units of measurement.


Example 1: 1,340 mL= _____ L

  • First, identify the unit conversion. You are converting a smaller unit to a larger unit. 1,000 mL = 1 L or 1,000 mL / 1 L
  • Create the proportion using the unit rate.
  • 1,000 mL / 1 L = 1,340 mL / x L
  • Then, divide 1,340 mL by 1,000 to determine what 1,000 mL was multiplied by to get 1,340 mL. 1,340 ÷ 1,000 = 1.34
  • 1,000 mL × 1.34 / 1 L × 1.34 = 1,340 mL / 1.34 L
  • 1,340 milliliters is equal to 1.34 liters.


Example 2: 66 g = _____ mg

  • First, identify the unit conversion. You are converting a larger unit to a smaller unit. 1,000 mg = 1 g or 1,000 mg / 1 g
  • 1,000 mg / 1 g = x g / 66 g
  • Since 1 grams was multiplied by 66 to get 66 grams, the 1,000 milligrams must also be multiplied by 66.
  • 1,000 mg × 66 / 1 g × 66 = 6,600 mg / 66 g
  • 66 grams is equal to 6,600 milligrams.


Example 3: 1,123 m = _____ km

  • First, identify the unit conversion. You are converting a smaller unit to a larger unit. 1,000 m = 1 km or 1,000 m / 1 km
  • 1,000 m / 1 km = 1,123 m / x km
  • 1,123 ÷ 1,000 = 1.123
  • 1,000 m × 1.123 / 1 km × 1.123 km = 1,123 m / 1.123 km
  • 1,123 meters is equal to 1.123 kilometers.
Converting units using Proportions and Ratios
Converting Measurements Using Proportions
Converting metric measurements with proportions
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Practice Converting within the Customary Measurement System Word Problems! Make sure you are logged into your Khan Academy!

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Practice Converting Metric Units! Make sure you are logged into your Khan Academy!

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Practice Converting within the Metric System Word Problems! Make sure you're logged into your Khan Academy!

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Read About Converting Between Measurement Systems

Sometimes it is necessary to convert between customary and metric units of measurements. These measurement conversions will be estimates, because you cannot make an exact measurement when converting between systems of measurement. Let’s look at an example.


Randy ran a 20 kilometer race. How many miles did Randy run?

  • First, set up a proportion.
  • 1.61 km / 1 mi = 20 km / x mi
  • Next, cross multiply.
  • 1.61 × x = 1 × 20
  • 1.61 × x = 20
  • Then, divide both sides by 1.61 to solve for x.
  • x × 1.61 ÷ 1.61 = x
  • 20 ÷ 1.61 = 12.42
  • The answer is 12.42.
  • Randy ran 12.42 miles.


Guided Practice

How many meters are in 67 feet?

  • First, set up a proportion.
  • 0.305 m / 1 ft. = x m / 67 ft.
  • 1 foot was multiplied by 67 to get 67 feet. So, you must do the same to 0.305 meters.
  • 0.305 m × 67 / 1 ft. × 67 = 20.435 m / 67 ft.
  • Therefore 67 feet is approximately 20.435 meters.


Examples


Example 1: John ran 5 kilometers. How many miles did he run?

  • First, set up a proportion.
  • 1.61 km / 1 mi = 5 km / x mi
  • Next, cross multiply.
  • 1.61 × x = 1 × 5
  • 1.61 × x = 5
  • Then, divide both sides by 1.61 to solve for x.
  • x × 1.61 ÷ 1.61 = x
  • 5 ÷ 1.61 = 3.11
  • The answer is 3.11 miles.
  • Therefore, 5 kilometers is approximately 3.11 miles.


Example 2: Kary measured out 12 inches on a ruler. About how many centimeters would that be?

  • First, set up a proportion.
  • 2.54 cm / 1 in. = x cm / 12 in.
  • 1 inch was multiplied by 12 to get 12 inches. You must multiply the 2.54 cm by 12 also to create an equivalent rate.
  • 2.54 cm × 12 / 1 in. × 12 = 30.48 cm / 12 in.
  • Therefore, 12 inches is 30.48 centimeters.


Example 3: Sandy ran 15 meters. About how many feet is that?

  • First, set up a proportion.
  • 0.305 m / 1 ft. = 15 m / x ft.
  • Next, cross multiply.
  • 0.305 × x = 1 × 15
  • 0.305 × x = 15
  • Then, divide both sides by 0.305 to solve for x.
  • x × 0.305 ÷ 0.305 = x
  • 15 ÷ 0.305 = 49.18
  • The answer is 49.18.
  • Therefore, 15 meters is approximately 49.18 feet.

Aleks- Converting between metric and US Customary Unit Systems
Converting Between the Customary and Metric Systems - Mrs. Renfro
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Practice Converting Between Customary and Metric Unit Systems!

Percents

Learning Target: I understand that a percent is a quantity compared to 100. I can convert between fractions decimals and percents. I can use proportional reasoning to problem solve with percents.


Essential Questions: What are percents? How can I convert between decimals, fractions, and percents? How can I solve problems involving percents?

Read about Percents

Percent means the number of parts per 100. You know that 100% of anything is the entire or whole amount. Therefore, percents can tell you what part of the whole you are dealing with. To understand percents, you will set up ratios.


A ratio is a comparison of two numbers and that a ratio can be written in three ways. For example: 1 to 2, 1:2, or 1/2. Percents are ratios in which the second number (or the denominator) is 100.


Let’s look at an example.

  • If there are 13 red jelly beans and 15 yellow jelly beans in a jar, the ratio of red jelly beans to yellow jelly beans can be written as 13 to 15, 13:15, or 13/15. Each of these ratios is read as “thirteen to fifteen.”


A percent is a type of ratio. You are now going to apply percents directly to ratios. A percent is a ratio that compares a number to 100. Percent means “per hundred” and the symbol for percent is %. 100% represents the ratio 100 to 100 or 1/1. Therefore, the value of 100% is 1.


Let’s look at another example.

  • If there are 100 jelly beans in a jar and 19 are black, you can say that 19/100 or 19% of the jelly beans in the jar are black.


Let’s look at one more situation that uses a real-life scenario.

  • There were 100 questions on a test and Amanda answered 92 of them correctly. What percent did she answer correctly? What percent did she answer incorrectly?
  • Amanda answered 92 out of 100 questions correctly. First, you can write this as the ratio 92/100.
  • Next, since the denominator is 100, you can write this ratio in percent form as 92%.
  • Then, since there were 100 questions on the test and Amanda answered 92 correctly, you can determine that she answered 8/100, or 8 out of 100 incorrectly.
  • Next, you can write this as the ratio 8/100 and as the percent 8%.
  • The answer is Amanda answered 92% of the questions correctly and 8% incorrectly.


Examples


Example 1: Solve the following problem. Karen ate 12 out of 100 blueberries. What percent of the blueberries did she eat?

  • First, you can write this as the ratio 12/100.
  • Then, since the denominator is 100, you can write this ratio in percent form as 12%.
  • The answer is she ate 12% of the blueberries.


Example 2: Solve the following problem. Joey answered 93 questions correctly out of 100 questions on his test. What percent of the questions did he answer correctly and what percent did he answer incorrectly?

  • First, you can write this as the ratio 93/100
  • Next, since the denominator is 100, you can write this ratio in percent form as 93%.
  • Then, since there were 100 questions on the test and Joey answered 93 correctly, you can determine that he answered 7/100, or 7 out of 100 incorrectly.
  • Next, you can write this as the ratio 7/100 and as the percent 7%.
  • The answer is Joey answered 93% of the questions correctly and 7% incorrectly.


Example 3: Solve the following problem. Sarah gathered 25 roses out of 100 flowers. What percent of the flowers were roses?

  • First, you can write this as the ratio 25/100.
  • Then, since the denominator is 100, you can write this ratio in percent form as 25%.
  • The answer is 25% of the flowers were roses.

Math Antics - What Are Percentages?
Colin Dodds - Percentages (Math Song)

Read About Converting Between Decimals and Percents

Decimals represent a part of a whole just like fractions and percents do. You are able to write fractions as percents and percents as fractions. You can also convert a decimal to a percent and a percent back to a decimal again. A decimal organizes numbers according to place value. When you write a decimal as a percent, you also keep track of the place value. A percent is a part of a whole out of 100. Well, so is a decimal.


Let’s see if you can make better sense of the connection between a decimal and a percent using the decimal 0.52.


This decimal states that you have 52 hundredths. The percent sign means “out of 100” so you have 52 hundredths or 52 parts out of 100. Once you see the connection, you can write this and all decimals as a percent. 0.52 = 52%. Both mean that you have 52 parts out of 100. You move the decimal point two places to the right to show hundredths. Percent is “out of 100,” so the % sign is the same as two decimal places: 0.52 = 52/100 = 52%.


Let’s look at how you write any percent as a decimal.

  • First, drop the % symbol.
  • Then, move the decimal point two places to the left.
  • Finally, add zeros to the right of the decimal point as placeholders, if necessary.


Let’s take a look at another example.

  • Write 9% as a decimal.
  • First, drop the % sign: 9
  • Next, move the decimal point two places to the left. Because 9 is only one digit, you have to add a placeholder zero to show the two decimal places.
  • 0.09
  • The answer is 9% written as a decimal is 0.09.


Let’s look at one final example.

  • Write 35.5% as a decimal.
  • First, drop the % sign: 35.5
  • Next, move the decimal point two places to the left.
  • 0.355
  • The answer is written as a decimal, 35.5% is 0.355.


Guided Practice


Write 25% as a decimal.

  • First, drop the % sign: 25
  • Although you can’t see it, the decimal point falls immediately to the right of any whole number. Next, move the decimal point two places to the left. This is the hundredths place. Remember that two places is hundredths. You just dropped the percent sign, which means out of 100, so your answer has to reflect “out of 100” in a different way.
  • 0.25
  • The answer is 25% written as a decimal is 0.25.


Examples


Example 1: What is 35% written as a decimal?

  • First, drop the % sign: 35
  • Next, move the decimal point two places to the left.
  • 0.35
  • The answer is 35% written as a decimal is 0.35.


Example 2: What is 2% written as a decimal?

  • First, drop the % sign: 2
  • Next, move the decimal point two places to the left. Because 2 is only one digit, you have to add a placeholder zero to show the two decimal places.
  • 0.02
  • The answer is 2% written as a decimal is 0.02.


Example 3: What is 18.7% written as a decimal?

First, drop the % sign.

18.7

Next, move the decimal point two places to the left.

0.187

The answer is 18.7% written as a decimal is 0.187.


You can write percents as decimals, and you can also write decimals as percents.

First, move the decimal point two places to the right. Add zeros to the right of the decimal point as placeholders, if necessary. Then, write a % symbol after the resulting number.


Let’s apply these steps to an example. Write 0.78 as a percent.


  • This decimal is already written in hundredths because there are two digits after the decimal point. All you have to do is move the decimal two places to the right and add a percent sign.
  • 0.78 = 78%
  • The answer is 0.78 written as a percent is 78%.


Let’s try another example. Write 0.345 as a percent.

  • This decimal is written in thousandths because there are three digits after the decimal point. You only need to move the decimal point two places to the right to make it a percent, so you will have a digit left over that remains after the decimal point.
  • 0.345 = 34.5%
  • The answer is 0.345 written as a percent is 34.5%.


Notice that sometimes you can have a decimal in a percent. In this case, it means that you have 34 and one-half percent. Not all percents are whole percents.


Let’s try one more example. Write 3.5 as a percent.

  • This decimal is written with a whole number and five tenths. You still move the decimal point to the right two places, adding a zero as a placeholder. Notice that because you have a whole number in front of the decimal point that the percent will be greater than 100.
  • 3.5 = 3.50 = 350%
  • The answer is 3.5 written as a percent is 350%.


Guided Practice


Write 0.6 as a percent.

  • This decimal is written in tenths because there is only one digit after the decimal point. When you move the decimal two places to the right, you will therefore need to add a zero placeholder.
  • 0.6 = 0.60 = 60%
  • The answer is 0.6 written as a percent is 60%.


Examples


Example 1: Write 0.45 as a percent.

  • This decimal is already written in hundredths because there are two digits after the decimal point. Therefore, all you have to do is move the decimal two places to the right and add a percent sign.
  • 0.45 = 45%
  • The answer is 0.45 written as a percent is 45%.


Example 2: Write 2.5 as a percent.

  • This decimal is written with a whole number and five tenths. You still move the decimal point to the right two places, adding a zero as a placeholder. Notice that because you have a whole number in front of the decimal point that the percent will be greater than 100.
  • 2.5 = 2.50 = 250%
  • The answer is 2.5 written as a percent is 250%.


Example 3: Write 0.875 as a percent.

  • This decimal is written in thousandths because there are three digits after the decimal point. You only need to move the decimal point two places to the right to make it a percent, so you will have a digit left over that remains after the decimal point.
  • 0.875 = 87.5%
  • The answer is 0.875 written as a percent is 87.5%.

Read about Converting Between Fractions and Percents

A fraction can be written as a percent if it has a denominator of 100. Sometimes, you will be given a fraction with a denominator of 100 and sometimes you will have to rewrite the fraction to have a denominator of 100 before you can write it as a percent.


For example: 9/100 This fraction is already written with a denominator of 100, so you can just change it to a percent. 9/100 = 9%


A proportion is two equal ratios. If a fraction does not have a denominator of 100, you can write a fraction equal to it that does have a denominator of 100 and then solve the proportion.


Let’s look at an example.

  • Write 3/5 as a percent.
  • First, notice that the denominator is not 100. Therefore, you need to create a new fraction equivalent to this one with a denominator of 100.
  • Next, set up a proportion.
  • 3/5 = x/100
  • Then, you can cross multiply to find the value of x.
  • 3/5 = x/100
  • 3 times 100 = 5 times x
  • 300 = 5 times x
  • 300 divided by 5 = x times 5 divided by 5
  • 60 = x
  • 3/5 = 60/100
  • Now you have a fraction with a denominator of 100, and you can write it as a percent.The answer is that the fraction is equal to 3/5 = 60%.


To work with an improper fraction, you have to think about what improper means. An improper fraction is greater than 1, so the percent would be greater than 100%. Sometimes you can have percents that are greater than 100%. Most often they are not, but it is important to understand how to work with a percent that is greater than 100%.


You already know some common fraction equivalents for percents. Think of 25 cents, 50 cents, and 75 cents. 25 cents means 25 cents out of a dollar, or 25% of a dollar. Since a quarter is 25 cents,

  • 1/4 = 25%.
  • 50 cents means 50 cents out of a dollar, or 50% of a dollar. Since a half dollar is 50 cents, 1/2 = 50%.
  • 75 cents means 75 cents out of a dollar, or 75% of a dollar. Since three quarters of a dollar is 75 cents, 3/4 = 75%.


Let’s look at an example with a fraction that doesn‘t convert easily to a percent.

  • Write 2/3 as a percent.
  • First, set up the proportion.
  • 2/3 = x/100
  • Next, cross multiply to solve for the value of x.
  • 3 times x = 2 times 100
  • 3 times x = 200
  • x times 3 divided by 3 = 200 divided by 3
  • x = 66.6...
  • Notice that you end up with a decimal and it is a repeating decimal. If you keep dividing, you will keep ending up with 6s. Therefore, you can leave this percent with one decimal place represented.
  • So 2/3 = 66.6/100
  • The answer is 66.6%.


Sometimes, you will see fractions like this, but you will get used to them and often you can learn the percent equivalents of these fractions by heart.


As a final example, let’s take a look at a real-life word problem.

  • James ate three out of ten pieces of pizza. What percent of the pizza did he eat? What percent didn’t he eat?
  • First, let’s write a fraction to show the part of the pizza that James did eat.
  • 3/10

  • Next, you convert that to a fraction out of 100 by setting up a proportion.
  • 3/10 = x/100
  • Then you can write it as a percent.
  • 3 times 100 = 10 times x
  • 300 = 10 times x
  • 300 divided by 10 = x times 10 divided by 10
  • 30 = x
  • So, 3/10 = 30/100 = 30%
  • The answer is James ate 30% of the pizza, and James did not eat 70% of the pizza.


Guided Practice


Write 9/4 as a percent.

  • First, you write a proportion with a denominator of 100.
  • 9/4 = x/100
  • Next, you cross multiply to find the value of x.
  • 9 times 100 = 4 times x
  • 900 = 4 times x
  • 900 divided by 4 = x times 4 divided by 4
  • 225 = x
  • So, 9/4 = 225/100 = 225%
  • The answer is 94 is equal to 225%.


Examples


Example 1: Write 1/4 as a percent.

  • First, set up the proportion.
  • 1/4 = x/100
  • Next, cross multiply to solve for the value of x.
  • 25 = x
  • 1/4 = 25/100
  • The answer is 25%.


Example 2: Write 25 as a percent.

  • First, set up the proportion.
  • 2/5 = x/100
  • Next, cross multiply to solve for the value of x.
  • 40 = x
  • 2/5 = 40/100
  • The answer is 40%.


Example 3: Write 4/40 as a percent.

  • First, set up the proportion.
  • 4/40 = x/100
  • Next, cross multiply to solve for the value of x.
  • 10 = x
  • 4/40 = 10/100
  • The answer is 10%.

Math Antics - Convert any Fraction to a Decimal
Crazy Conversions- a song about converting between fractions, decimals, and percents
Converting percent to decimal and fraction | Decimals | Pre-Algebra | Khan Academy
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Practice Converting Fractions, Decimals, and Percents! Make sure you are logged into your Khan Academy!

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Practice Converting Fractions, Decimals, and Percents! Make sure you are logged into your Khan Academy!

Comparing Fractions, Decimals, and Percents

To compare and order rational numbers, you should first convert each number to the same form so that they are easier to compare. Usually it will be easier to convert each number to a decimal. Then you can use a number line to help you order the numbers. To order the rational numbers from greatest to least, or least to greatest, you need to use inequality signs. The inequality signs are:

  • Greater than: >
  • Less than: <
  • Greater than or equal to: ≥
  • Less than or equal to: ≤


Let’s look at an example. Place the following number in order from least to greatest: 1/8, 8%, and 0.8.

  • First, convert 1/8 and 8% into decimals.
  • 1/8 = 0.125
  • 8% = 0.08
  • Next, place the three decimals on a number line between 0 and 1. Then, since you have compared the numbers by placing them on a number line, you can order the rational numbers.
  • The answer is 8% < 1/8 < 0.8.


Here is another example. Which inequality symbol correctly compares 0.29% to 0.029?

  • First, convert the percent into a decimal.
  • 0.29% = 0.0029
  • Next, compare 0.0029 to 0.029.
  • 0.0029 < 0.029
  • So, 0.29% < 0.029 .


Remember, the key to comparing and ordering rational numbers is to be sure that they are all in the same form. You want to have all fractions, all decimals or all percentages so that your comparisons are accurate. You may need to convert before you compare!!


Guided Practice


Order the following rational numbers from least to greatest: 0.5%, 0.68 , 3/15

  • First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation, let’s use percent.
  • Since 0.5% is in the percent form, it is done.
  • 0.68 = 68%
  • 3/15 = 20%
  • Next, order the numbers. 0.5% < 20% < 68%. Before writing your final answer be sure to write them as they first appeared.
  • So the answer is 0.5% < 3/15 < 0.68.


Examples


Example 1: Which inequality symbol correctly compares 0.56 to 4/5?

  • First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation, let’s use fractions.
  • Since 4/5 is in the fraction form, it is done.
  • 0.56 = 56/100 = 14/25
  • Next, get a common denominator so you can compare the fractions.
  • 4/5 = 16/25
  • 0.56 = 14/25
  • Then, order the numbers. Be sure to write them as they first appeared. 0.56 < 4/5
  • The answer is 0.56<45.


Example 2: Which inequality symbol correctly compares 0.008 to 0.8%?

  • First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation, let’s use percent.
  • Since 0.8% is in the percent form, it is done.
  • 0.008 = 0.8%
  • Next, order the numbers. Be sure to write them as they first appeared. 0.008 = 0.8%
  • The answer is 0.008=0.8%.
Comparing and Ordering Fractions, Decimals, and Percents
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Practice Comparing Fractions, Decimals, and Percents

Solving Percent Problems

Learning Target: I can solve real world percent problems using proportional reasoning.


Essential Question: How can you use proportional reasoning to solve real world percent problems?

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Read About Solving Percent Problems to Find a Part

In previous lessons, you were shown how to convert a decimal to a percent and a percent to a decimal. Thus, if you were asked to Find 15% of 120, you would multiply 0.15 by 120, to get an answer of 18. But what would you do if you given this problem: 8 is what percent of 20? In this problem, the percent is the unknown quantity! We need to figure out how to find this unknown quantity.


Every statement of percent can be expressed verbally as: "One number is some percent of another number." Percent statements will always involve three numbers. For example:

____ is ____ % of ____.


In the problem, 8 is what percent of 20?, the number 8 is some percent of the number 20. Looking at this problem, it is clear that 8 is the part and 20 is the whole. Similarly, in the statement, "One number is some percent of another number.", the phrase "one number" represents the part and "another number" represents the whole. Thus the statement, "One number is some percent of another number," can be rewritten:

  • "One number is some percent of another number" becomes, "The part is some percent of the whole."

From previous lessons we know that the word "is" means equals and the word "of" means multiply. Thus, we can rewrite the statement above:


The statement:"The part is some percent of the whole.", becomes the equation:

  • the part = some percent x the whole


Since a percent is a whose second term is 100, we can use this fact to rewrite the equation above as follows:

  • the part = some percent x the whole becomes:
  • the part = percent/100 x the whole


Dividing both sides by "the whole" we get the following proportion:

  • PART / WHOLE = PERCENT / 100


Since percent statements always involve three numbers, given any two of these numbers, we can find the third using the proportion above. Let's look at an example of this.


Problem 1: If 8 out of 20 students in a class are boys, what percent of the class is made up of boys?

  • Analysis: In this problem, you are being asked 8 is what percent of 20? You are given two numbers from the proportion above and asked to find the third. The percent is the unknown quantity in this problem. We need to find this unknown quantity.
  • Identify: The phrase 8 is means that 8 is the part.
  • The phrase what percent tells us that percent is the unknown quantity. This unknown quantity will be represented by x in our proportion.
  • The phrase of 20 means that 20 is the whole.
  • Substitute: Now we can substitute these values into our proportion.
  • PART / WHOLE = PERCENT / 100 becomes 8/20 = x/100
  • Solve: Cross multiply and we get: 20x = 800
  • Divide both sides by 20 to solve for x and we get: x = 40
  • Solution: 8 is 40% of 20. Therefore, 40% of the class is made up of boys.


Note that in Problem 1 we did not have to cross multiply to solve the proportion. We could have used equivalent fractions instead (i.e., since 20 multiplied by 5 equals 100, we get that 8 multiplied by 5 equals x, so x equals 40).


In Problem 1 we were asked 8 is what percent of 20? and we found the solution by substituting into a proportion. But how would we solve this problem: 18 is 40% of what number? and how would we solve this problem: What is 20% of 45? We will look at these last two problems below.


Problem 2: 18 is 40% of what number?

  • Identify: The phrase 18 is means that 18 is the part.
  • 40% means that 40 will replace percent in our proportion.
  • The phrase of what number represents the whole and is the unknown quantity. We will let variable x represent this unknown quantity in our proportion.
  • Substitute: Now we can substitute these values into our proportion.
  • PART / WHOLE = PERCENT / 100 becomes 18/x = 40/100
  • Solve: Cross multiply and we get: 40x = 18(100) or 40x = 1800
  • Divide both sides by 40 to solve for x and we get: x = 45
  • Solution: 18 is 40% of 45


Problem 3: What is 20% of 45?
  • Identify: The phrase what is means represents the part and is the unknown quantity. We will let variable x represent this unknown quantity in our proportion.
  • 20% means that 20 will replace percent in our proportion.
  • The phrase of 45 means that 45 is the whole.
  • Substitute: Now we can substitute these values into our proportion.
  • PART / WHOLE = PERCENT / 100 becomes x/45 = 20/100
  • Solve: Cross multiply and we get: 100x = 45(20) or 100x = 900
  • Divide both sides by 100 to solve for x and we get: x = 9
  • Solution: 9 is 20% of 45


In Problems 1, 2 and 3 we are given two numbers and asked to find the third by using a proportion. However, the unknown quantity was different for each problem. Let's compare these problems.


In Problem 1 we let x represent the unknown quantity "what percent"; in Problem 2 we let x represent the unknown quantity "of what number"; and in Problem 3 we let x represent the unknown quantity "What is." Thus, we solved three different percent problems, where in each problem, two numbers were given and we were asked to find the third. We did this by letting a variable represent the unknown quantity and then substituting the given values into a proportion to solve for the unknown quantity.


Note that in all three percent statements, the whole always follows the word "of" and the part always precedes the word "is." This is not surprising since our original statement is, "One number is some percent of another number." Thus, we can revise our proportion as follows:

  • PART / WHOLE = PERCENT / 100 becomes IS (part) / OF (whole) = PERCENT / 100


Let's solve some more percent problems using proportions.


Problem 4: What is 25% of 52?

  • Identify: 25% means that 25 will replace PERCENT in our proportion.
  • 52 is the whole and will replace OF in our proportion.
  • The part is the unknown quantity and will be represented by p in our proportion.
  • Substitute: Now we can substitute these values into our proportion.
  • IS (part) / OF (whole) = PERCENT / 100 becomes p/52 = 25/100
  • Solve: Cross multiply and we get: 100p = 52(25) or 100p = 1300
  • Divide both sides by 100 to solve for p and we get: p = 13
  • Solution: 13 is 25% of 52


Note that we could restate this problem as, "Find 25% of 52", and get the same answer. However, in the interest of consistency, we will use proportions to solve percent problems throughout this lesson. In Problems 5 through 7, we will use n to represent the unknown quantity.


Problem 5: What percent of 56 is 14?

  • Identify: 56 is the whole and will replace OF in our proportion.
  • 14 is the part and will replace IS in our proportion.
  • PERCENT is the unknown quantity in our proportion, to be represented by n.
  • Substitute: IS (part) / OF (whole) = PERCENT / 100 becomes 14/56 = n/100
  • Solve: Cross multiply and we get: 56n = 14(100), or 56n = 1400
  • Divide both sides by 56 and we get: n = 25
  • Solution: 25% of 56 is 14


Problem 6: 18 is 75% of what number?
  • Identify: 18 is the part and will replace IS in our proportion.
  • 75% means that 75 will replace PERCENT in our proportion.
  • The whole is the unknown quantity in our proportion, to be represented by n.
  • Substitute: IS (part) / OF (whole) = PERCENT / 100 becomes 18/n = 75/100
  • Solve: Cross multiply and we get: 75n = 18(100) or 75n = 1800
  • Divide both sides by 75 and we get: n = 24
  • Solution: 18 is 75% of 24


Problem 7: What is 15% of 200?

  • Identify: 15% means that 25 will replace PERCENT in our proportion.
  • 200 is the whole and will replace OF in our proportion.
  • The part is the unknown quantity in our proportion, to be represented by n
  • Substitute: IS (part) / OF (whole) = PERCENT / 100 becomes n/200 = 15/100
  • Solve: Cross multiply and we get: 100n = 200(15) or 100n = 3000
  • Divide both sides by 100 and we get: n = 30
  • Solution: 30 is 15% of 200


Now that we have solved a number of percent problems using proportions, we can go back to the type of problem presented at the beginning of this lesson. In Problems 8 through 10 we will solve real world problems, using different variables to represent the unknown quantity in each problem.


Problem 8: At Little Rock School, 476 students ride their bike to school. If this number is 85% of the school enrollment, then how many students are enrolled?

  • Identify: This problem can be rewritten as 476 is 85% of what number?
  • 476 is the part and will replace IS in our proportion.
  • The percent given is 85%.
  • The whole is the unknown quantity, so y will represent the OF in our proportion.
  • Substitute: IS / OF = PERCENT / 100 becomes 476 / y = 85 / 100 Solve:Cross multiply and we get: 85y = 47600
  • Divide both sides by 85 and we get: y = 560
  • Solution: There are 560 students enrolled at Little Rock School.


Problem 9: A football team won 75% of 120 games in a season. How many games is that?

  • Identify: This problem can be rewritten as What is 75% of 120?
  • 120 is the whole and will replace the OF in our proportion.
  • The percent given is 75%.
  • The part is the unknown quantity, so p will represent the IS in our proportion.
  • Substitute: IS / OF = PERCENT / 100 becomes p/120 = 75/100
  • Solve: Cross multiply and we get: 100p = 9000
  • Divide both sides by 85 and we get: p = 90
  • Solution: The team won 90 games.


Problem 10: Jennie has $300 and she spends $15. What percent of her money is spent?

  • Identify: This problem can be rewritten as $15 is what percent of $300?
  • 15 is the part and will replace the IS in our proportion.
  • 300 is the whole and will replace the OF in our proportion.
  • Percent is the unknown quantity, so x will represent the PERCENT in our proportion.
  • Substitute: IS / OF = PERCENT / 100 becomes 15/300 = x/100
  • Solve: Cross multiply and we get: 300x = 1500
  • Divide both sides by 300 and we get: x = 5
  • Solution: Jennie spent 5% of her money.


Summary: Every statement of percent can be expressed verbally as: "One number is some percent of another number." Percent statements will always involve three numbers. Given two of these numbers, we can find the third by substituting into one of the proportions below.

  • PART / WHOLE = PERCENT / 100 OR IS (part) / OF (whole) = PERCENT / 100


In this lesson, we solved percent problems using proportions by following this procedure:

  1. Read the percent problem.
  2. Identify what information is given.
  3. Identify what information is unknown.
  4. Use a variable to represent the unknown quantity.
  5. Set up a proportion for the problem by substituting the given information and the variable into one of the proportions listed above.
  6. Evaluate and solve the proportion in Step 5 to find the unknown quantity.
https://www.youtube.com/watch?v=FaDtge_vkbg
https://www.youtube.com/watch?v=-lUEWEEpmIo
https://www.youtube.com/watch?v=N4kDzoQOngY
https://www.youtube.com/watch?v=DAikW24_O0A
https://www.youtube.com/watch?v=d1oNF88SAgg
https://www.youtube.com/watch?v=AjYil74WrVo
https://www.youtube.com/watch?v=gKywkLHV6Ko
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