Ratios

Question 1

Considering the Total Number of Items in a Ratio

Answer 1

Because ratios compare the amounts of two different things, the total number of items is a multiple of the sum of the numbers in the ratio.

So, for example, if the ratio is 3 dogs for every 2 cats (3:2 or 3/2), then the total number of dogs has to be 3, 6, 9, or any multiple of 3. Same with the cats: the number has to be a multiple of 2 (e.g. 2, 4, 6, etc.)

You first need to find the sum of the numbers in the ratio: 3+2 = 5. So the total number of animals has to be a multiple of 5, with 3 for the dogs and 2 for the cats.

Question 2

Finding the Amount of a Specific Term of a Ratio

Answer 2

4 Steps:

1) Add the numbers in the ratio. (The total number of items is a multiple of this sum)

2) Divide the total number of items by that sum.

3) Multiply that quotient by each term in the ratio.

4) Add the answers to double-check that they add up to the total. (Be sure to do this step to check your work)

Question 3

Setting Up a Proportion (When Problem Features an Existing Ratio, and You Have to Change the Number of Items While Maintaining the Ratio)

Answer 3

1) Set up the existing ratio as a fraction.

2) Set up the new additions as a fraction, with x as the unknown value.

3) Set the fractions equal to each other.

4) Solve for x.

Question 4

Combining Two or More Ratios

Answer 4

Example Question:

Sam’s jazz shop has 6 saxophones for every 5 drum kits and 2 drum kits for every 3 trombones. What’s the ratio of saxophones to trombones?

1) Set up the ratios as A:B.

Place the item that the ratios have in common (drum kits) into a column.

Saxes Drums Trombones

6 : 5

2 : 3

2) Find a common multiple for the item that these ratios have in common. [In this instance, both ratios include drum kits. The least common multiple of 5 and 2 (the #’s of drum kits) is 10]

3) Multiply each term in the ratios so that the quantity of the item in common equals the common multiple (from Step 2). Multiply both terms of each ratio by the same number, as though you’re getting a common denominator. Here, you want the number of drum kits to equal 10. Multiply both terms in the first ratio by 2, and multiply both terms in the second ratio by 5:

Saxes Drums Trombones

6(2) : 5(2)

2(5) : 3(5)

Saxes Drums Trombones

12 : 10

10 : 15

12 : 10 : 15

4) Write out a combined ratio. The combined ratio of saxophones to drum kits to trombones is 12:10:15. To answer the question, give only the ratio of sacophones to trombones, which is 12:15, or 4:5.