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Prealgebra / LEVEL 8 · DIFFICULTY 3/5

Percents & Changing Baselines

Find parts, recover wholes and reason about successive changes.

3 stages · 24 practice problems · two 6-question assessment forms

STAGE 8.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Percent, Part & Whole

Useful preparation: Ratios & Multiway Shares · Fractions & Terminating Decimals

Goal: Translate a percent statement into a relationship between three quantities.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Percent means per hundred

A percent is a dimensionless fraction with denominator 100. Convert to a fraction or decimal before multiplying.

p%=p100\displaystyle p\% = \frac{p}{100}

Identify the baseline whole

The same part can be a different percent of different wholes. Mark which quantity is called 100 percent.

part = rate × whole

Recover a missing quantity

To find the whole, divide the known part by the decimal rate. To find the rate, divide part by whole, then express per hundred.

whole = part / rate
WORKED EXAMPLE 1

18 is 15%\displaystyle 15\% of what number?

  1. Write 18=0.15 W\displaystyle 18 = 0.15\, W.
  2. Divide by 0.15.

120

WORKED EXAMPLE 2

What percent of 80 is 28?

  1. Compute 2880=0.35\displaystyle \frac{28}{80} = 0.35.
  2. Convert to percent by multiplying by 100.

35%\displaystyle 35\%

Common pitfalls

Possible mix-up: 20%\displaystyle 20\% means multiply by 20.

Use 20100=0.2\displaystyle \frac{20}{100} = 0.2.

Possible mix-up: Divide the whole by the part to find the percent.

Use part divided by whole.

Explain it to yourself

Can a percent exceed 100? Give a meaningful example.

Preview the eight practice prompts
  1. Problem 1

    Find 35%\displaystyle 35\% of 240.

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  2. Problem 2

    What percent of 60 is 27? Enter the number before the percent sign.

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  3. Problem 3

    24 is 16%\displaystyle 16\% of what number?

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  4. Problem 4

    Write 0.625 as a percent. Enter the number before the percent sign.

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  5. Problem 5

    Find 125%\displaystyle 125\% of 48.

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  6. Problem 6

    A quantity is 38\displaystyle \frac{3}{8} of its whole. What percent is it? Enter the number before the percent sign.

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  7. Problem 7New context

    A library lends 42 books, which is 28%\displaystyle 28\% of its science collection. How many science books are in the collection?

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  8. Problem 8New context

    A battery initially stores 80 units. It uses 18 units. What percent of its initial charge remains? Enter the number before the percent sign.

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Open stage 8.1 in the student workspace →

STAGE 8.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Percent Increase, Decrease & Reverse Change

Useful preparation: Percent, Part & Whole

Goal: Measure a change relative to its original baseline.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A change rate uses the starting amount

Subtract original from new and divide by original. The result is positive for an increase and negative for a decrease.

change rate = (new − original)/original

Use a multiplier for the new amount

An increase adds a fraction of the original; a decrease subtracts that fraction.

new = original × (1 ± rate)

Reverse the multiplier

Recover an original price from a discounted or increased price by dividing by the multiplier used, not by applying the opposite percent.

original = discounted price / (1 − discount rate)
WORKED EXAMPLE 1

A fee rises from $45 to $54. Find the percent increase.

  1. The increase is 9 dollars.
  2. Divide by the original 45.

20%\displaystyle 20\%

WORKED EXAMPLE 2

After a 30%\displaystyle a\, 30\% discount, a jacket costs $63. Find its original price.

  1. The discounted price is 70%\displaystyle 70\% of the original.
  2. 630.70=90\displaystyle \frac{63}{0.70} = 90.

90 dollars

Common pitfalls

Possible mix-up: Use the final value as the denominator for percent change.

The standard change rate uses the initial value.

Possible mix-up: Undo a 20%\displaystyle a\, 20\% discount by adding 20%\displaystyle 20\% to the discounted price.

Divide by 0.8 to recover the original baseline.

Explain it to yourself

Why is a rise from 40 to 50 not the same percent as a fall from 50 to 40?

Preview the eight practice prompts
  1. Problem 1

    Increase 160 by 15%\displaystyle 15\%.

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  2. Problem 2

    Decrease 250 by 12%\displaystyle 12\%.

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  3. Problem 3

    A value rises from 80 to 104. What is the percent increase? Enter only the numerical percent.

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  4. Problem 4

    A value drops from 90 to 72. What is the percent decrease? Enter a positive numerical percent.

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  5. Problem 5

    After a 25%\displaystyle a\, 25\% discount, a bag costs $54. What was its original price in dollars?

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  6. Problem 6

    After an 8%\displaystyle 8\% increase, a fee is $81. What was the old fee in dollars?

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  7. Problem 7New context

    A garden harvested 140 kg last year and 175 kg this year. What is the percent increase? Enter only the numerical percent.

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  8. Problem 8New context

    A device loses 18%\displaystyle 18\% of its value and is now worth $328. What was its previous value in dollars?

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Open stage 8.2 in the student workspace →

STAGE 8.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Successive Changes & Percentage Points

Useful preparation: Percent Increase, Decrease & Reverse Change

Goal: Multiply changing baselines and distinguish two types of percent comparison.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Each change acts on its current baseline

For repeated increases or decreases, multiply the factors. Adding the rates ignores the effect of the first change on the second baseline.

final = start × factor₁ × factor₂

Equal opposite rates do not cancel

An increase by r followed by a decrease by r multiplies by 1−r2\displaystyle 1 - r^{2}, which is below one for positive r.

(1+r) (1−r)=1−r2\displaystyle \left(1 + r\right)\, \left(1 - r\right) = 1 - r^{2}

Percentage points compare reported percents

A change from 30%\displaystyle 30\% to 36%\displaystyle 36\% is six percentage points. Relative to 30%\displaystyle 30\%, it is a 20%\displaystyle a\, 20\% increase. State which comparison is intended.

point change = new percent − old percent
WORKED EXAMPLE 1

A $100 price is cut 20%\displaystyle 20\%, then increased 10%\displaystyle 10\%. Find the final price.

  1. After the cut it is $80.
  2. The increase is 10%\displaystyle 10\% of $80, giving $88.

88 dollars

WORKED EXAMPLE 2

A success rate moves from 40%\displaystyle 40\% to 50%\displaystyle 50\%. Find the percentage-point increase.

  1. Subtract the two numerical percents.
  2. 50−40=10\displaystyle 50 - 40 = 10.

10 percentage points

Common pitfalls

Possible mix-up: A 10%\displaystyle A\, 10\% increase and 10%\displaystyle 10\% decrease cancel.

Their combined factor is 1.1×0.9=0.99\displaystyle 1.1 \times 0.9 = 0.99.

Possible mix-up: Percentage points and relative percent change mean the same thing.

One subtracts percentages; the other divides the change by the initial percentage.

Explain it to yourself

Under what conditions does changing the order of two percentage multipliers preserve the result?

Preview the eight practice prompts
  1. Problem 1

    A $200 price rises 10%\displaystyle 10\%, then rises another 10%\displaystyle 10\%. What is the final price?

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  2. Problem 2

    A $150 price falls 20%\displaystyle 20\%, then rises 20%\displaystyle 20\%. What is the final price?

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  3. Problem 3

    A $80 item is discounted 25%\displaystyle 25\%, then taxed 5%\displaystyle 5\% on the sale price. What is the total in dollars?

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  4. Problem 4

    A rate changes from 32%\displaystyle 32\% to 47%\displaystyle 47\%. How many percentage points is the increase?

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  5. Problem 5

    A rate rises from 20%\displaystyle 20\% to 25%\displaystyle 25\%. What is the relative percent increase? Enter only the numerical percent.

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  6. Problem 6

    A value is cut 10%\displaystyle 10\% twice. What total percent decrease results? Enter a positive numerical percent.

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  7. Problem 7New context

    A club grows 25%\displaystyle 25\% in spring and 20%\displaystyle 20\% in autumn, ending with 90 members. How many members did it start with?

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  8. Problem 8New context

    A store offers either 30%\displaystyle 30\% off, or 20%\displaystyle 20\% off followed by 15%\displaystyle 15\% off the reduced price. On a $100 item, how many dollars cheaper is the two-discount offer?

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Open stage 8.3 in the student workspace →
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