Lesson 12

Solving More Ratio Problems

Let’s compare all our strategies for solving ratio problems.

12.1: You Tell the Story

Describe a situation with two quantities that this tape diagram could represent.

A tape diagram showing 7 parts and 3 parts.

 

12.2: A Trip to the Aquarium

Consider the problem: A teacher is planning a class trip to the aquarium. The aquarium requires 2 chaperones for every 15 students. The teacher plans accordingly and orders a total of 85 tickets. How many tickets are for chaperones, and how many are for students?

  1. Solve this problem in one of three ways:

    Use a triple number line.

    A triple number line, one each for kids, chaperones, and total.

    Use a table.
    (Fill rows as needed.) 

    kids chaperones total
    15 2 17

    Use a tape diagram.

    A tape diagram. For "kids" 15 equal parts and for "chaperones" 2 equal parts. A brace from "kids" to "chaperones" is labeled 85.
  2. After your class discusses all three strategies, which do you prefer for this problem and why?


Use the digits 1 through 9 to create three equivalent ratios. Use each digit only one time.

\(\boxed{\phantom{3}}:\boxed{\phantom{3}}\) is equivalent to \(\boxed{\phantom{3}}\,\boxed{\phantom{3}}:\boxed{\phantom{3}}\) and \(\boxed{\phantom{3}}\,\boxed{\phantom{3}}:\boxed{\phantom{3}}\,\boxed{\phantom{3}}\)

12.3: Salad Dressing and Moving Boxes

Solve each problem, and show your thinking. Organize it so it can be followed by others. If you get stuck, consider drawing a double number line, table, or tape diagram.

  1. A recipe for salad dressing calls for 4 parts oil for every 3 parts vinegar. How much oil should you use to make a total of 28 teaspoons of dressing?
  2. Andre and Han are moving boxes. Andre can move 4 boxes every half hour. Han can move 5 boxes every half hour. How long will it take Andre and Han to move all 72 boxes?

Summary

When solving a problem involving equivalent ratios, it is often helpful to use a diagram. Any diagram is fine as long as it correctly shows the mathematics and you can explain it.

Let’s compare three different ways to solve the same problem: The ratio of adults to kids in a school is \(2:7\). If there is a total of 180 people, how many of them are adults?

  • Tape diagrams are especially useful for this type of problem because both parts of the ratio have the same units (“number of people") and we can see the total number of parts.

    A tape diagram. For "number of adults" 2 equal parts and for "number of kids" 7 equal parts.

    This tape diagram has 9 equal parts, and they need to represent 180 people total. That means each part represents \(180 \div 9\), or 20 people.

    Two tape diagrams, one for number of adults and one for number of kids.

    Two parts of the tape diagram represent adults. There are 40 adults in the school because \(2\boldcdot 20 = 40\).

  • Double or triple number lines are useful when we want to see how far apart the numbers are from one another. They are harder to use with very big or very small numbers, but they could support our reasoning.

    A triple number line, one each for adults, kids, and total.

  • Tables are especially useful when the problem has very large or very small numbers.

    A table with three columns for kids, adults, and total.

    We ask ourselves, “9 times what is 180?” The answer is 20. Next, we multiply 2 by 20 to get the total number of adults in the school.

Another reason to make diagrams is to communicate our thinking to others. Here are some good habits when making diagrams:

  • Label each part of the diagram with what it represents.
  • Label important amounts.
  • Make sure you read what the question is asking and answer it.
  • Make sure you make the answer easy to find.
  • Include units in your answer. For example, write “4 cups” instead of just “4.”

  • Double check that your ratio language is correct and matches your diagram.

Glossary Entries

  • double number line diagram

    A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. The locations of the tick marks match on both number lines. The tick marks labeled 0 line up, but the other numbers are usually different.

    A double number line for teaspoons of red paint: 0, 3, 6, 9, 12 and teaspoons of yellow paint: 0, 5, 10, 15, 20.
  • per

    The word per means “for each.” For example, if the price is $5 per ticket, that means you will pay $5 for each ticket. Buying 4 tickets would cost $20, because \(4 \boldcdot 5 = 20\).

  • same rate

    We use the words same rate to describe two situations that have equivalent ratios.

    For example, a sink is filling with water at a rate of 2 gallons per minute. If a tub is also filling with water at a rate of 2 gallons per minute, then the sink and the tub are filling at the same rate.

  • table

    A table organizes information into horizontal rows and vertical columns. The first row or column usually tells what the numbers represent.

    For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.

    pet tail length (inches)
    dog 22
    cat 12
    mouse 2
  • tape diagram

    A tape diagram is a group of rectangles put together to represent a relationship between quantities.

    For example, this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint.

    tape diagrams

    If each rectangle were labeled 5, instead of 10, then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint.