Ratios

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Ratios, rates and proportions

Ratios, rates and proportions on the SAT test

SAT Subscore: Problem solving and data analysis

This subject includes 3 connected topics that deal with representing relationships using division: ratios, rates and proportions.

Since ratios, rates and proportions are written as fractions, before learning this topic you should learn solving fractions topic.

A ratio is a comparison of two numbers, represented by a division of their amounts. The ratio between a and b can be represented using a colon as a:b or as a fraction a/b. For example: The number of girls in the class is 22 and the number of boys in the class is 11. The ratio of boys to girls in the class is 11:22, reducing 11:22 gives 1:2. The ratio of girls to boys is 22:11, reducing 22:11 gives 2:1=2.

Equivalent ratios are ratios that express the same relationship between numbers. Two ratios are equivalent if we can reduce or expand one ratio and get the other ratio.

Complementary ratios are ratios that add up to a whole that is 1.

There are 2 types of ratios:  a part to part ratio and a part to whole ratio.

A rate is a quantity compared to another related quantity, where the quantities have different units. For example: The speed per hour is the rate measured by the number of miles per a unit of time of 1 hour.

A unit rate is a rate compared to a single unit quantity (the denominator is 1).

To calculate the unit rate, divide the total of one quantity (the numerator) by the number of units of the other quantity (the denominator).

 A proportion is an equality between 2 or more equivalent ratios. The proportion between a, b, c and d can be represented using a colon as a:b=c:d or as a fraction a/b=c/d. For example: A carrot cake recipe is composed of 3 cups of flour and 1 cup of sugar, if we put 3 cups of sugar in the bowl, we need to add 9 cups of flour: 3/1=9/3 or 3:1=9:3

We can write proportions in 2 ways: There are same units above the divisor line and below the division line or there are same units at the nominator and the denominator of each ratio.

Solving proportions is done using the cross product method which involves multiplying the numerator of one fraction by the denominator of another fraction and then equaling the multiplications.

Continue reading this page for detailed explanations and examples.

Ratios

A ratio is a comparison of two numbers, represented by a division of their amounts. The ratio between a and b can be represented using a colon as a:b or as a fraction a/b.

Reducing and expanding ratios: Since ratios are represented as fractions, ratios can be reduced or expanded. To expend a ratio, we multiply the numbers in the ratio by a same factor. To reduce a ratio, we divide the numbers in the ratio by a same factor.
For example: The number of girls in the class is 22 and the number of boys in the class is 11. The ratio of boys to girls in the class is 11:22, reducing 11:22 gives 1:2 (we divided by 11). The ratio of girls to boys is 22:11, reducing 22:11 gives 2:1=2 (we divided by 11).

Equivalent ratios and complementary ratios

Equivalent ratios are ratios that express the same relationship between numbers. Two ratios are equivalent if we can reduce or expand one ratio and get the other ratio.
For example: the ratios 11:22 and 1:2 are equivalent ratios, since we can multiply the ratio 1:2 by 11 (expending) and get the ratio 11:22 or we can divide the ratio 11:22 by 11 (reducing) and get the ratio 1:2.

Complementary ratios are ratios that add up to a whole that is 1. Note that, to get this some of 1, we need to include all the ratios in the group.
In the previous example, if we add the number of boys to the number of pupils that is 1/3 and the number of girls to the number of pupils that is 2/3 we get 1 (the class includes only boys and girls, therefore the ratios must add up to 1).

Types of ratios: a part to part ratio and a part to whole ratio

A part to part ratio compares the amount of one quantity against the amount of other quantity (each one of the amounts is a part from a total that is not included in the ratio calculation).
In the previous example: the number of boys to the number of girls 11:22=1:2 is a part to part ratio. The number of girls to the number of boys 22:11=2:1=2 is also a part to part ratio.

A part to whole ratio compares the amount of one quantity against the total amount.
In the previous example: the total number of pupils in the class is 11+22=33, therefore a part to whole ratio will be the number of boys to the number of pupils 11:33=1:3 or the number of girls to the number of pupils 22:33=2:3.

Calculating amounts of quantities

We may need to calculate the amount of quantity before we can calculate its ratio. This can be done only if we know the total amount and the amounts of all other complementary ratios.

Consider the following example:

A carrot cake recipe is composed of 3 cups of flour, 1 cup of sugar and oil.

If the total amount of the cake mixture is 5 cups, calculate the ratio of the oil to the flour and the ratio of the oil to the cake.

First, we need to find the amount of oil in one cake. We know that the mixture amount of one cake is 5 cups and that besides oil it includes 3 cups of flour and 1 cup of sugar, therefore the amount of oil in 1 cake is 5-3-1=1 cup.

Calculating the ratio of the oil to the flour: one cake includes 1 cup of oil and 3 cups of flour, therefore the ratio of the oil to the flour is 1:3.

Calculating the ratio of the oil to the cake: one cake includes 1 cup of oil and it contains 5 cups of mixture, therefore the ratio of the oil to the cake is 1:5.

Consider the following example:

One glass of milkshake includes 2 scoops of ice cream (one scoop is equal to 1/2 cup), 3/4 cups of milk and a syrup.

If the capacity of one glass is 2 cups, what is the ratio of syrup to milkshake?   

Translating 2 scoops of ice cream to glasses: 1 scoop=1/2 cup therefore 2 scoops of ice cream=2*1/2=1 cups of ice cream. Finding how much syrup we need to put in 1 glass of milkshake: 1 glass of milkshake=2 cups= 1 cups of ice cream + 3/4 cups of milk + x syrup. 2=1+3/4+x 2=13/4+x x=2-13/4=1/4 the amount of syrup in a glass of milkshake is 1/4 cup. The ratio of syrup to milkshake is 1/4 cup to 2 cups, that is (1/4)/2=1/8.

Finding a part to part ratio given complementary ratios

Since all complementary ratios add up to 1, we can calculate any part to part ratio given the other part to part ratios.

Consider the following example:

The ratio of staff to the total number of the people on the cruise ship is 1:5. What is the ratio of the tourists to the total number of people on a cruise ship?

Since the only people on the cruise ship are staff and tourists, we can calculate the ratio of the tourists to the total number of people on a cruise ship. We know that the unknown tourists to total ratio x and staff to total ratio 1:5 are complementary ratios, therefore they add up to 1:
1/5+x=1
x=1-1/5=4/5
The ratio of tourists to the total the people on a cruise ship is 4/5 (or 4:5).

Consider the previous example

One glass of milkshake includes 2 scoops of ice cream (one scoop is equal to 1/2 cup), 3/4 cups of milk and a syrup.

If the capacity of one glass is 2 cups, what is the ratio of syrup to milkshake?

Translating 2 scoops of ice cream to glasses:
1 scoop=1/2 cup therefore 2 scoops of ice cream=2*1/2=1 cups of ice cream.

Finding the ratio of syrup to milkshake using complementary ratios:
The ratio of milk to milkshake is 3/4 glass to 2 glasses, that is (3/4):2=3/8.
The ratio of ice cream to milkshake is 1 glass to 2 glasses, that is 1/2.
The ratio of syrup to milkshake is 1-3/8-1/2=8/8-3/8-4/8=1/8.

Calculating part to part ratios using part to whole ratios

We can calculate a part to part ratio by division of the corresponding part to whole ratios.

Consider the previous example:

One glass of milkshake includes 2 scoops of ice cream (one scoop is equal to 1/2 cup), 3/4 cups of milk and a syrup.

If the capacity of one glass is 2 cups, what is the ratio of syrup to milk?

We found earlier that:
1 scoop=1/2 cup therefore 2 scoops of ice cream=2*1/2=1 cups of ice cream.
The ratio of milk to milkshake is 3/4 glass to 2 glasses, that is (3/4):2=3/8.
The ratio of syrup to milkshake is 1-3/8-1/2=8/8-3/8-4/8=1/8.

The ratio of syrup to milk using the ratios to the whole is:
(1/8)/(3/8)= (1/8)*(8/3)=(1*8)/(8*3)=8/24=1/3

Checking the answer by calculating with the amounts from the total:
(1/4 cup)/(3/4 cup)= (1/4)*(4/3)=(1*4)/(4*3)=4/12=1/3

Calculating amounts of quantities using part to whole ratio

We can calculate amount of quantity by multiplying the total amount by the part to whole ratio.

Consider the following example:

The ratio of staff to the total number of people on the cruise ship is 1:5.

What is the number of the tourists if the total number of people on the cruise ship is 5,000?

The number of staff is the total amount 5,000 multiplied by the part to total ratio of staff 1:5, this gives us 5000*1/5=1000.
The number of tourists is the total amount 5,000 minus the number off staff 1,000, this gives us 5,000-1,000=4,000.

Another way is to calculate the part to total ratio of the tourists to the staff first, this is done by using the complementary ratios:
The ratio of staff to the total number of people on the cruise ship 1:5 and the ratio of tourists to the total number of people on the cruise ship x are equal to 1 (these ratios are complementary ratios, since there are no other people on the cruise ship).
1/5+x=1
x=1-1/5=4/5 the ratio of tourists to the total number of people is 4/5 or 4:5

The number of tourists is the total amount 5,000 multiplied by the part to total ratio of tourists 4:5, this gives us 5,000*4/5=4,000.

Calculating the total amount using ratios

We can calculate the total amount by dividing an amount of quantity by its part to total ratio.

Consider the following example:

The ratio of staff to the total number of people on the cruise ship 1:6.

If there are 500 of staff on the ship, what is the total number of people on the ship?  

500/(1/6)=500*6/1=3,000 the total number of people on the ship is 3,000.

This example can also be solved using proportions, as explained below.

Rates

A rate is a quantity compared to another related quantity, where the quantities have different units.

A unit rate is a rate compared to a single unit quantity (the denominator is 1). For example:
The speed per hour is the rate measured by the number of miles per a unit of time of 1 hour.
The price is the rate measured by the amount of dollars per 1 unit of product.
The heart rate is measured by the number of contractions of the heart per a unit of time of 1 minute.

To calculate the unit rate, divide the total of one quantity (the numerator) by the number of units of the other quantity (the denominator). Consider the following examples:

If the group walked 4 miles in 2 hours, what is the speed of the group (miles fer hour)?

 

The speed rate is the total of one quantity (the number of miles which is 4) divided by the number of units of the other quantity (the number of hours which is 2). The speed is therefore 4/2=2/1=2 miles per 1 hour or 2 miles per hour.

Note that we can also calculate the unit rate using proportion, this calculation takes longer therefore it is less recommended:
4 miles / 2 hours = X miles / 1 hour
4/2=x
x=2

The price of 4 books is 88 dollars, the price of 3 notebooks is 9 dollars. What is the price of 2 books and 2 notebooks?

First, we need to calculate the rates, which are prices in this example:
The price of 1 book= the price of the books 88 (the total of one quantity)/the number of books 4 (the number of units of the other quantity)= 88/4= 22 dollars.
The price of 1 notebook= the price of the notebooks 9 (the total of one quantity)/the number of notebooks 3 (the number of units of the other quantity)= 9/3= 3 dollars.

The price of 2 books and 2 notebooks is 2*22+2*3=44+6=50 dollars.

Proportions

A proportion is an equality between 2 or more equivalent ratios. The proportion between a, b, c and d can be represented using a colon as a:b=c:d or as a fraction a/b=c/d.

The ways of representing proportions

We can write proportions in 2 ways, note that the way of writing doesn’t change the equality of the proportion.

There are same units above the divisor line and below the division line.

For example: A carrot cake recipe is composed of 3 cups of flour and 1 cup of sugar, if we put 3 cups of sugar in the bowl, what is the amount of flour we need to add?

Writing a flour above the divisor line and a sugar below the divisor line:

3 cups of flour / 1 cups of sugar = x cups of flour / 3 cups of sugar

3/1=x/3

x=9 if there are 3 cups of sugar in the bowl, we need to add 9 cups of flour

Writing a sugar above the divisor line and a flour below the divisor line:

1 cups of sugar / 3 cups of flour = 3 cups of sugar / x cups of flour

1/3=3/x

x=9 if there are 3 cups of sugar in the bowl, we need to add 9 cups of flour

There are same units at the nominator and the denominator of each ratio.

For example: A carrot cake recipe is composed of 3 cups of flour and 1 cup of sugar, if we put 3 cups of sugar in the bowl, what is the amount of flour we need to add?

One ratio includes flour and the other sugar:

3 cups of flour / x cups of flour = 1 cups of sugar / 3 cups of sugar

3/x=1/3

x=9 if there are 3 cups of sugar in the bowl, we need to add 9 cups of flour

We can also replace the numerators and the denominators:

x cups of flour/3 cups of flour = 3 cups of sugar/1 cups of sugar

x/3=3/1

x=9 if there are 3 cups of sugar in the bowl, we need to add 9 cups of flour