4.2.4: More and Less than 1 Percent

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Illustrative Mathematics
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Lesson

Let's explore percentages smaller than 1%.

Exercise $\PageIndex{1}$: Number Talk: What Percentage?

Determine the percentage mentally.

10 is what percentage of 50?

5 is what percentage of 50?

1 is what percentage of 50?

17 is what percentage of 50?

Exercise $\PageIndex{2}$: Waiting Tables

During one waiter’s shift, he delivered appetizers, entrées, and desserts. What percentage of the dishes were desserts? appetizers? entrées? What do your percentages add up to?

Exercise $\PageIndex{3}$: Fractions of a Percent

1. Find each percentage of 60. What do you notice about your answers?

$30\%\text{ of }60\qquad 3\%\text{ of }60\qquad 0.3\%\text{ of }60\qquad 0.03\%\text{ of }60$

2. 20% of 5,000 is 1,000 and 21% of 5,000 is 1,050. Find each percentage of 5,000 and be prepared to explain your reasoning. If you get stuck, consider using the double number line diagram.

1% of 5,000
0.1% of 5,000
20.1% of 5,000
20.4% of 5,000

Figure $\PageIndex{1}$: A double number line with 12 tick marks. The first tick mark is followed by a break and then 11 evenly spaced tick marks. For the top number line, the number 0 is on the first tick mark, 1000 on the second, and 1050 on the twelfth. For the bottom number line, the percentage 0% is on the first tick mark, 20% on the second, and 21% on the twelfth.

3. 15% of 80 is 12 and 16% of 80 is 12.8. Find each percentage of 80 and be prepared to explain your reasoning.

15.1% of 80
15.7% of 80

Are you ready for more?

To make Sierpinski's triangle,

Start with an equilateral triangle. This is step 1.
Connect the midpoints of every side, and remove the middle triangle, leaving three smaller triangles. This is step 2.
Do the same to each of the remaining triangles. This is step 3.
Keep repeating this process.

Figure $\PageIndex{2}$: Step 1, blue triangle. Step 2, triangle broken into 4 equal sized triangles, 1 white and 3 blue. Step 3, triangle broken into 12 small triangles, 9 blue and 3 white and one larger white triangle in center.

What percentage of the area of the original triangle is left after step 2? Step 3? Step 10?
At which step does the percentage first fall below 1%?

Exercise $\PageIndex{4}$: Population Growth

The population of City A was approximately 243,000 people, and it increased by 8% in one year. What was the new population?
The population of city B was approximately 7,150,000, and it increased by 0.8% in one year. What was the new population?

Summary

A percentage, such as 30%, is a rate per 100. To find 30% of a quantity, we multiply it by $30\div 100$, or 0.3.

The same method works for percentages that are not whole numbers, like 7.8% or 2.5%. In the square, 2.5% of the area is shaded.

To find 2.5% of a quantity, we multiply it by $2.5\div 100$, or 0.025. For example, to calculate 2.5% interest on a bank balance of $80, we multiply $(0.025)\cdot 80=2$, so the interest is $2.

We can sometimes find percentages like 2.5% mentally by using convenient whole number percents. For example, 25% of 80 is one fourth of 80, which is 20. Since 2.5 is one tenth of 25, we know that 2.5% of 80 is one tenth of 20, which is 2.

Glossary Entries

Definition: Percentage Decrease

A percentage decrease tells how much a quantity went down, expressed as a percentage of the starting amount.

For example, a store had 64 hats in stock on Friday. They had 48 hats left on Saturday. The amount went down by 16.

This was a 25% decrease, because 16 is 25% of 64.

Definition: Percentage Increase

A percentage increase tell how much a quantity went up, expressed as a percentage of the starting amount.

For example, Elena had $50 in the bank on Monday. She had $56 on Tuesday. The amount went up by $6.

This was a 12% increase, because 6 is 12% of 50.

Practice

Exercise $\PageIndex{5}$

The student government snack shop sold 32 items this week. For each snack type, what percentage of all snacks sold were of that type?

Table $\PageIndex{1}$
snack type	number of items sold
fruit cup	$8$
veggie sticks	$6$
chips	$14$
water	$4$

Exercise $\PageIndex{6}$

Select all the options that have the same value as $3\frac{1}{2}\%$ of 20.

$3.5\%$ of $20$
$3\frac{1}{2}\cdot 20$
$(0.35)\cdot 20$
$(0.035)\cdot 20$
$7\%$ of $10$

Exercise $\PageIndex{7}$

22% of 65 is 14.3. What is 22.6% of 65? Explain your reasoning.

Exercise $\PageIndex{8}$

A bakery used 30% more sugar this month than last month. If the bakery used 560 kilograms of sugar last month, how much did it use this month?

(From Unit 4.2.2)

Exercise $\PageIndex{9}$

Match each situation to a diagram. The diagrams can be used more than once.

Figure $\PageIndex{6}$: Tape diagram A. Both tapes same size. Last year, entire tape labeled 100 percent. This year, small blue portion of tape labeled 15 percent. Tape diagram B. Last year, entire tape labeled 100 percent. This year, small blue portion of tape labeled 15 percent and extends farther than last year tape.

The amount of apples this year decreased by 15% compared with last year's amount.
The amount of pears this year is 85% of last year's amount.
The amount of cherries this year increased by 15% compared with last year's amount.
The amount of oranges this year is 115% of last year's amount.

Diagram A
Diagram B

(From Unit 4.2.1)

Exercise $\PageIndex{10}$

A certain type of car has room for 4 passengers.

Write an equation relating the number of cars ($n$) to the number of passengers ($p$).
How many passengers could fit in 78 cars?
How many cars would be needed to fit 78 passengers?

(From Unit 2.2.3)

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snack type	number of items sold
fruit cup	\(8\)
veggie sticks	\(6\)
chips	\(14\)
water	\(4\)