RATIO PROBLEMS

RATE (advanced)

Concept: A ratio is a comparison of two numbers. It can be written in several ways. For example, when comparing the numbers 1 and 2, we can express the comparison as 1 to 2, or 1 : 2 or ½.

Always write the terms of the ratio in the same order as the problem compares them.

Just as we can reduce fractions to lowest terms, we can simplify ratios to their lowest terms, although a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.

Many ratio problems require more than one step.

You may not be given both of the numbers that you need to write a ratio. In this case, you may need to solve another equation to find the missing number.

Finally, when two ratios are written as equal ratios, the equation is called a proportion. For example, is a proportion. In many proportion problems, one of the four terms is missing.

The proportion can be solved using the Cross-product Rule.

Cross-product Rule
To find the missing number in a proportion, cross multiply, and divide the product by the third number. For example, in the proportion , first multiply . Then divide . In this problem .

Steps for Solving Simple Ratio Problems

Step 1 | Set up the ratio, writing the terms in the same order that the problem compares the terms.
Step 2 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.)

Steps for Solving Multiple Step Ratio Problems

Step 1 | Set up the first ratio, writing the terms in the same order that the problem compares the terms.
Step 2 | Set up an equation to solve for the unknown number needed for the ratio.
Step 3 | Set up a ratio, writing the terms in the same order that the problem compares the terms.
Step 4 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.)

Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
Step 2 | Apply the Cross-product Rule to solve for X.

Steps for Solving Multistep Proportion Problems

Step 1 | Set up the first ratio using given information.
Step 2 | Set up an equation, if necessary, to solve for any unknown information.
Step 3 | Set up another ratio, if necessary, using the new information.
Step 4 | Form a proportion using two ratios. Use the ratio from either Step 1 or Step 3 for the first ratio in the proportion, and another ratio, set up in the same order, while substituting X for the unknown number.
Step 5 | Apply the Cross-product Rule to solve for X.

Example 1: The ratio of ballerinas in azurite to ballerinas in malachite was 59 to 17. Those who wore azurite numbered 1016 more than 3 times the number who wore malachite. How many wore malachite? Round the answer to the nearest whole number.

Use the Steps for Solving Multistep Proportion Problems

Step 1 | Set up the first ratio using the given information. |
Step 2 | Set up an equation, if necessary, to solve for any unknown information. |
| We already know the ratio of the ballerinas that wore azurite to those that wore malachite. Now we need to know how many wore malachite based on the total number of those that wore azurite. We are given that those who wore azurite were 1016 more than 3 times those who wore malachite. Writing this out in equation form we get:
azurite = 1016 + (3 x malachite)
Now we have enough information to set up another ratio using this new information. |
Step 3 | Set up another ratio, if necessary, using the new information. |
| We can take the ratio in Step 1 and plug in the equation that we developed in Step 2 to create a new ratio: Since we are solving for those who wore malachite, then we can replace the word malachite with the variable x. |
Step 4 | Form a proportion using two ratios. Use the ratio from either Step 1 or Step 3 for the first ratio in the proportion, and another ratio, set up in the same order, while substituting X for the unknown number. |
| This step is not necessary |
Step 5 | Apply the Cross-product Rule to solve for X. |
| To find the missing number in a proportion, cross multiply, 17 x (1016 + 3 x) = 59 x and then multiply out the left side of the equation. 17272 + 51 x = 59 x Combine like terms and solve for x, 17272 = 51x – 59x 17272 = 8x 2159 = x Therefore, our answer for X is 2159. |
| The number of ballerinas that wore malachite was 2159. |

Example 2: A wildfire rumor traveled 93 kilometers in 12 parsecs. How long does it take to travel 161 kilometers going the same rate? Round the answer to the nearest tenth.

Use the Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step 2 | Apply the Cross-product Rule to solve for X. |
| X is equal to 20.77. The problem is asking us to round the number to the nearest tenth, therefore our answer is going to be 20.8 |
| To travel 161 kilometers it will take the wildfire rumor 20.8 parsecs. |

Example 3: In the bag were ebony and teal socks. If the ratio of ebony to teal socks was 2:15, what ratio of the socks were ebony?

Use the Steps for Solving Simple Ratio Problems

Step 1 | Set up the ratio, writing the terms in the same order that the problem compares the terms. |
| Let’s write out the ratio so we can see it in understandable form: 2 ebony socks to 15 teal socks Now we add up all the socks, 2 + 15 = 17 and since the problem is asking us what ratio of the socks were ebony, we place the number of ebony socks first followed by the total amount of socks: 2 : 17 |
Step 4 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.) Cannot be simplified. |
| The ratio of ebony to teal (ebony: teal) is 2:17. |

Example 4: Roger made the 275 mile trip in 7 hours less than it took Gracie. This was because he traveled 4 times as fast as Gracie. How long did it take Gracie? Round answer to the nearest hundredth.

Use the Steps for Solving Multiple Step Ratio Problems

Step 1 | Set up the first ratio, writing the terms in the same order that the problem compares the terms. |
| This step is not necessary |
Step 2 | Set up an equation to solve for the unknown number needed for the ratio. |
| We have four bits of information given to us in the problem. Since 4 of them deal with time and the question is asking us how long it took Gracie, we can discard the 275miles. If we let R represent Roger and G represent Gracie, then our first ratio would be: (1) R = G – 7 because it took Roger 7 hours less than it took Gracie to make the trip. The other bit of information given to us is that Roger did it because he had traveled 4 times as fast as Gracie. In equation form that would be: (2) 4 x R = G Let’s rewrite (2) so that R is by itself as it is in (1): (3) R = ¼ G |
Step 3 | Set up a ratio, writing the terms in the same order that the problem compares the terms. |
| Now we can set up our ratio using (1) and (3) from the previous step and solve. R = G – 7 = ¼ G G –¼ G = 7 ¾ G =7 G = 7 (4/3) G = (7 x 4) ÷ 3 G = 9.33 It took Gracie 9.33 hours. |
Step 4 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.) This step is not necessary |
| To travel 275 miles it took Gracie 9.33 hours. |