Lesson 16Finding the Percentage

Let’s find percentages in general.

Learning Targets:

  • I can solve different problems like “60 is what percentage of 40?” by dividing and multiplying.

16.1 True or False: Percentages

Is each statement true or false? Be prepared to explain your reasoning.

  1. 25% of 512 is equal to \frac14 \boldcdot 500 .
  2. 90% of 133 is equal to (0.9) \boldcdot 133 .
  3. 30% of 44 is equal to 3% of 440.
  4. The percentage 21 is of 28 is equal to the percentage 30 is of 40.

16.2 Jumping Rope

A school held a jump-roping contest. Diego jumped rope for 20 minutes.

  1. Jada jumped rope for 15 minutes. What percentage of Diego’s time is that?

  2. Lin jumped rope for 24 minutes. What percentage of Diego’s time is that?

  3. Noah jumped rope for 9 minutes. What percentage of Diego’s time is that?

  4. Record your answers in this table. Write the quotients in the last column as decimals.
    time (minutes) percentage \text{time} \div 20
    Diego 20 100 \frac{20}{20} = 1\text{   }
    Jada 15 \frac{15}{20} = \text{          }
    Lin 24 \frac{24}{20} = \text{          }
    Noah 9 \frac{9}{20} = \text{          }
  5. What do you notice about the numbers in the last two columns of the table?

16.3 Restaurant Capacity

A restaurant has a sign by the front door that says, “Maximum occupancy: 75 people.” Answer each question and explain or show your reasoning.

  1. What percentage of its capacity is 9 people?
  2. What percentage of its capacity is 51 people?
  3. What percentage of its capacity is 84 people?

Are you ready for more?

Water makes up about 71% of the Earth’s surface, while the other 29% consists of continents and islands. 96% of all the Earth’s water is contained within the oceans as salt water, while the remaining 4% is fresh water located in lakes, rivers, glaciers, and the polar ice caps.

If the total volume of water on Earth is 1,386 million cubic kilometers, what is the volume of salt water? What is the volume of fresh water?

Lesson 16 Summary

What percentage of 90 kg is 36 kg? One way to solve this problem is to first find what percentage 1 kg is of 90, and then multiply by 36.

A table with two columns. The first column is labeled "mass in kilograms". The second column is labeled "percentage". The data are as follows: row 1: 90 kilograms, 100 percent; row 2: one kilogram, the fraction 1 over 90, end fraction, times 100; row 3: 36 kilograms, the fraction 36 over 90, end fraction, times 100. Arrows on both sides of the table from row 1 to row 2 are labeled "multiply by the fraction 1 over 90." Arrows on both sides of the table from row 2 to row 3 are labeled "multiply by 36."

From the table we can see that 1 kg is  \left(\frac{1}{90}\boldcdot 100\right) \% , so 36 kg is \left(\frac{36}{90}\boldcdot 100\right) \% or 40% of 90. We can confirm this on a double number line:

A double number line with eleven evenly spaced tick marks. The top number line is labeled “mass in kiograms" andd starting with the first tick mark 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 are labeled. The bottom number line is not labeled and starting with the first tick mark zero percent, 10 percent, 20 percent, 30 percent, 40 percent, 50 percent, 60 percent, 70 percent, 80, percent, 90 percent, 100 percent are labeled.

In general, to find what percentage a number C is of another number  B is to calculate  \frac{C}{B} of 100%. We can find do that by multiplying: \frac{C}{B}\boldcdot 100

Suppose a school club has raised $88 for a project but needs a total of $160. What percentage of its goal has the club raised? 

To find what percentage $88 is of $160, we find \frac {88}{160} of 100% or \frac {88}{160} \boldcdot 100 , which equals  \frac {11}{20} \boldcdot 100 or 55. The club has raised 55% of its goal. 

Lesson 16 Practice Problems

  1. A sign in front of a roller coaster says "You must be 40 inches tall to ride." What percentage of this height is:

    1. 34 inches?
    2. 54 inches?
  2. At a hardware store, a tool set normally costs $80. During a sale this week, the tool set costs $12 less than usual. What percentage of the usual price is the savings? Explain or show your reasoning.

  3. A bathtub can hold 80 gallons of water. The faucet flows at a rate of 4 gallons per minute. What percentage of the tub will be filled after 6 minutes?

  4. The sale price of every item in a store is 85% of its usual price.

    1. The usual price of a backpack is $30, what is its sale price?
    2. The usual price of a sweatshirt is $18, what is its sale price?
    3. The usual price of a soccer ball is $24.80, what is its sale price?
  5. A shopper needs 48 hot dogs. The store sells identical hot dogs in 2 differently sized packages. They sell a six-pack of hot dogs for $2.10, and an eight-pack of hot dogs for $3.12. Should the shopper buy 8 six-packs, or 6 eight-packs? Explain your reasoning.

  6. Elena is 56 inches tall.

    1. What is her height in centimeters? (Note: 100 inches = 254 centimeters)

    2. What is her height in meters?