Lesson 1.5 · Unit 1 · N-Q.1-3

How much more, and more than what

Every raise, price change and investment return is a percent change, and every percent change has a base. Most of the confusion in personal finance is a confusion about the base. This lesson fixes the rule once so that it does not need to be rediscovered in every unit that follows.

The method
  1. Percent change is the change divided by the original, times 100: \( \dfrac{\text{new} - \text{old}}{\text{old}} \times 100 \). The original is always the denominator.
  2. A positive result is an increase and a negative result is a decrease. From 40 to 46 the change is 6 and the percent change is 15 percent.
  3. To apply a percent change, multiply by one plus the rate. A 4 percent raise on \$52,000 is \( 52000 \times 1.04 = 54080 \). A 20 percent decrease multiplies by 0.80.
  4. To undo a percent change, divide by the same factor. The salary before a 4 percent raise to \$54,080 is \( \dfrac{54080}{1.04} = 52000 \).
  5. Successive changes multiply, they do not add. Raises of 3 percent and then 5 percent give \( 1.03 \times 1.05 = 1.0815 \), a total of 8.15 percent, not 8.
  6. An increase and the same percent decrease do not cancel. A 20 percent cut followed by a 20 percent raise leaves \( 0.80 \times 1.20 = 0.96 \) of the original, because the second percent is taken of a smaller amount.
  7. A percentage point is not a percent. From 5 percent to 7 percent is a rise of 2 percentage points and a 40 percent increase in the rate.
  8. Compare offers in the same period, in dollars first and percent second. A percent hides the size of the amount it is taken from.

Where students lose marks: dividing by the wrong number. The percent change from 40 to 46 is 6 divided by 40, not by 46. Dividing by the new value gives 13 percent instead of 15, and the error is worst exactly where the change is large.

Worked example

The problem. An analyst earns \$46,800. She is offered a 5 percent raise to stay, and she has another offer for \$49,400. (a) Find her salary after the raise. (b) Find the percent increase of the outside offer over her current pay. (c) After the 5 percent raise she receives a 3 percent raise the next year. Find her salary then and the total percent increase. (d) Explain why the total is not 8 percent.

Step one: name the givens. The current salary is \$46,800, which is the base for both comparisons. The raise is 5 percent. The outside offer is \$49,400.

Step two: the raise for (a). \( 46800 \times 1.05 = 49{,}140 \). Her salary becomes \$49,140.

Step three: the outside offer for (b). The change is \( 49400 - 46800 = 2600 \), and \( \dfrac{2600}{46800} \times 100 = 5.56 \) percent.

Step four: compare. The outside offer is \$260 a year higher than the raise (\$49,400 against \$49,140), which is a small difference once benefits and the move are considered.

Step five: the second raise for (c). The next raise is taken on the new salary: \( 49140 \times 1.03 = 50{,}614.20 \).

Step six: the total increase. \( \dfrac{50614.20 - 46800}{46800} \times 100 = 8.15 \) percent.

Step seven: explain for (d). The factors multiply: \( 1.05 \times 1.03 = 1.0815 \). The second raise is 3 percent of the larger salary, so it adds more than 3 percent of the original. The extra 0.15 percent is 3 percent of the first 5 percent.

Step eight: state the answers. After the raise, \$49,140. The outside offer is 5.56 percent higher than her current pay. After both raises, \$50,614.20, a total of 8.15 percent. A raise should be compared in dollars as well as percent, since 5 percent of a large salary is a bigger gift than 5 percent of a small one.

Practice · 10 problems

Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.

  1. A salary of \$52,000 rises 4 percent. Find the new salary.
    Show the full solution

    \( 52000 \times 1.04 = 54080 \). \$54,080

  2. Find the percent change when an hourly wage goes from \$40 to \$46.
    Show the full solution

    \( \dfrac{46 - 40}{40} \times 100 = 15 \). 15 percent increase

  3. A price of \$100 is cut by 30 percent and then raised by 30 percent. Find the final price.
    Show the full solution

    \( 100 \times 0.70 = 70 \), then \( 70 \times 1.30 = 91 \). The price is lower than the start because the second 30 percent is taken of \$70. \$91.00

  4. After a 4 percent raise a salary is \$54,080. Find the salary before the raise.
    Show the full solution

    Divide by the factor: \( \dfrac{54080}{1.04} = 52000 \). \$52,000

  5. A monthly rent falls from \$25 to \$20 in a worked example of a discount. Find the percent change.
    Show the full solution

    \( \dfrac{20 - 25}{25} \times 100 = -20 \). 20 percent decrease

  6. An employee gets raises of 3 percent, 4 percent and 2 percent in three years. Find the total percent increase.
    Show the full solution

    \( 1.03 \times 1.04 \times 1.02 = 1.092624 \), a total increase of 9.26 percent. Adding the three rates would give 9 percent, which understates the growth. 9.26 percent

  7. A job pays \$27.50 an hour for 40 hours and 52 weeks. Another pays \$56,000 a year. Find the annual pay of the first and the difference.
    Show the full solution

    \( 27.50 \times 2080 = 57{,}200 \). The difference is \( 57200 - 56000 = 1200 \) in favor of the hourly job. \$57,200; \$1,200 more

  8. A price is reduced by 20 percent and then increased by 20 percent. Find the final price from \$100, and find what percent increase would restore the original price after the cut.
    Show the full solution

    \( 100 \times 0.80 \times 1.20 = 96 \). To restore \$100 from \$80, the increase is \( \dfrac{100 - 80}{80} \times 100 = 25 \) percent. The restoring percent is larger than the cut because it is taken of a smaller amount. \$96.00; a 25 percent increase

  9. A rate rises from 5 percent to 7 percent. State the change in percentage points and in percent.
    Show the full solution

    The rise is \( 7 - 5 = 2 \) percentage points. As a percent change, the increase is \( \dfrac{7 - 5}{5} \times 100 = 40 \) percent of the original rate. 2 percentage points; a 40 percent increase

  10. A worker gets a 4 percent raise in a year when prices rise 3 percent. Find her real raise, meaning the change in what her pay buys.
    Show the full solution

    Pay is multiplied by 1.04 and prices by 1.03, so buying power is multiplied by \( \dfrac{1.04}{1.03} = 1.00971 \), a gain of about 0.97 percent. Subtracting the rates to get 1 percent is close but not exact. About 0.97 percent

← 1.4 Payroll taxes: Social Security and Medicare1.6 Self-employment and the gig economy →

Keep going in the full Financial Algebra course

All of U1 (Unit 1: Earning and Paychecks) is free to everyone, with no account. The course has 75 lessons, each with worked examples and practice problems that come with full solutions. A free account can also start a free month of the whole course, with no card.

Open this lesson in the courseSee the whole course