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Lesson 12Navigating a Table of Equivalent Ratios

Let’s use a table of equivalent ratios like a pro.

Learning Targets:

  • I can solve problems about situations happening at the same rate by using a table and finding a “1” row.
  • I can use a table of equivalent ratios to solve problems about unit price.

12.1 Number Talk: Multiplying by a Unit Fraction

Find the product mentally.

\frac13\boldcdot 21

\frac16 \boldcdot 21

(5.6) \boldcdot \frac18

\frac14\boldcdot (5.6)

12.2 Comparing Taco Prices

number of tacos price in dollars

Use the table to help you solve these problems. Explain or show your reasoning.

  1. Noah bought 4 tacos and paid $6. At this rate, how many tacos could he buy for $15?
  2. Jada’s family bought 50 tacos for a party and paid $72. Were Jada’s tacos the same price as Noah’s tacos?

12.3 Hourly Wages

Lin is paid $90 for 5 hours of work. She used the following table to calculate how much she would be paid at this rate for 8 hours of work.

A 2-column table with 5 rows of data. The first column is labeled "amount earned, in dollars" and the second column is labeled "time worked, in hours." Row 1: 90, 5; Row 2: 18, 1; Row 3: 144, 8; Row 4: blank, 3; Row 5: blank, 2.1. An arrow pointing from row 1 to row 2 is labeled "times one-fifth" and another arrow pointing from row 2 to row 3 is labeled "times 8."
  1. What is the meaning of the 18 that appears in the table?
  2. Why was the number \frac15 used as a multiplier?
  1. Explain how Lin used this table to solve the problem.

  2. At this rate, how much would Lin be paid for 3 hours of work? For 2.1 hours of work?

12.4 Zeno’s Memory Card

In 2016, 128 gigabytes (GB) of portable computer memory cost $32.

  1. Here is a double number line that represents the situation:

    A double number line: For "memory, in gigabytes" the numbers 0, 64, and 128 are indicated. For "cost, in dollars" the numbers 0, 16, and 32 are indicated.

    One set of tick marks has already been drawn to show the result of multiplying 128 and 32 each by \frac12 . Label the amount of memory and the cost for these tick marks.

    Next, keep multiplying by \frac12 and drawing and labeling new tick marks, until you can no longer clearly label each new tick mark with a number.

  2. Here is a table that represents the situation. Find the cost of 1 gigabyte. You can use as many rows as you need.

memory (gigabytes) cost (dollars)
128 32
  1. Did you prefer the double number line or the table for solving this problem? Why?

Are you ready for more?

A kilometer is 1,000 meters because kilo is a prefix that means 1,000. The prefix mega means 1,000,000 and giga (as in gigabyte) means 1,000,000,000. One byte is the amount of memory needed to store one letter of the alphabet. About how many of each of the following would fit on a 1-gigabyte flash drive?

  1. letters
  1. pages
  1. books
  1. movies
  1. songs

Lesson 12 Summary

Finding a row containing a “1” is often a good way to work with tables of equivalent ratios. For example, the price for 4 lbs of granola is $5. At that rate, what would be the price for 62 lbs of granola?

Here are tables showing two different approaches to solving this problem. Both of these approaches are correct. However, one approach is more efficient.

  • Less efficient

    A 2-column table with 6 rows of data. The first column is labeled "granola, in pounds" and the second column is labeled "price, in dollars." Row 1: 4, 5; Row 2: 8, 10; Row 3: 16, 20; Row 4: 32, 40; Row 5: 64, 80; Row 6: 62, 77.50. An arrow pointing from row 1 to row 2, then 2 to 3, then 3 to 4, then 4 to 5 is labeled "times 2". The last arrow from row 5 to 6 is labeled "subtract 2 pounds" on the left of the table and is labeled " subtract "$2.50" on the right.

  • More efficient

    A 2-column table with 3 rows of data. The first column is labeled "granola, in pounds" and the second column is labeled " price, in dollars." Row 1: 4, 5; Row 2: 1, 1.25; Row 3: 62, 77.50. An arrow pointing from row 1 to row 2 is labeled "times one-fourth" and another arrow pointing from row 2 to row 3 is labeled "times 62."

Notice how the more efficient approach starts by finding the price for 1 lb of granola.

Remember that dividing by a whole number is the same as multiplying by a unit fraction. In this example, we can divide by 4 or multiply by \frac14 to find the unit price.

Lesson 12 Practice Problems

  1. Priya collected 2,400 grams of pennies in a fundraiser. Each penny has a mass of 2.5 grams. How much money did Priya raise? If you get stuck, consider using the table. 

    number of pennies mass in grams
    1 2.5
    2,400

  2. Kiran reads 5 pages in 20 minutes. He spends the same amount of time per page. How long will it take him to read 11 pages? If you get stuck, consider using the table.

    time in minutes number of pages
    20 5
    1
    11

  3. Mai is making personal pizzas. For 4 pizzas, she uses 10 ounces of cheese.

    number of pizzas ounces of cheese
    4 10

    a. How much cheese does Mai use per pizza?

    b. At this rate, how much cheese will she need to make 15 pizzas?

  4. Clare is paid $90 for 5 hours of work. At this rate, how many seconds does it take for her to earn 25 cents?

  5. A car that travels 20 miles in \frac12 hour at constant speed is traveling at the same speed as a car that travels 30 miles in \frac34 hour at a constant speed. Explain or show why.

  6. Lin makes her favorite juice blend by mixing cranberry juice with apple juice in the ratio shown on the double number line. Complete the diagram to show smaller and larger batches that would taste the same as Lin's favorite blend.

    Lin makes her favorite juice blend by mixing cranberry juice with apple juice in the ratio shown on the double number line.

  7. Each of these is a pair of equivalent ratios. For each pair, explain why they are equivalent ratios or draw a representation that shows why they are equivalent ratios.

    1. 600:450 and 60:45
    2. 60:45 and 4:3
    3. 600:450 and 4:3