SAT · Problem-Solving and Data Analysis · TOPIC LESSON
Identify the base, use multipliers, and distinguish relative change from percentage points.
BY THE END, YOU CAN
BEFORE YOU START
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01 / UNDERSTAND & APPLY
A percentage p% is the fraction p/100 of a specified base. The relationship part = rate × whole becomes A = (p/100)B. Before calculating, name the base B: “20% of B” and “B is 20% of A” are different statements. Percentages can exceed 100 when the compared quantity is larger than the base.
The question may ask for the part, the whole or the rate. Solve the same relationship for the requested quantity instead of memorizing unrelated formulas.
02 / UNDERSTAND & APPLY
To find a part, multiply by the decimal rate. To find the base, divide the part by the decimal rate. To find a percentage, divide part by base and multiply by 100. The denominator must be the specified reference quantity.
ORIGINAL WORKED EXAMPLE
A class has 18 students who bike to school, representing 30% of the class. How many students are in the class?
Answer 60 students
03 / UNDERSTAND & APPLY
An increase of r as a decimal changes B to B + rB = (1 + r)B. A decrease changes it to B − rB = (1 − r)B. A 12% increase uses 1.12; a 12% decrease uses 0.88. The percent change from old to new is (new − old)/old × 100%.
The old value is the base for a change, even if the new value is larger or easier to divide by.
ORIGINAL WORKED EXAMPLE
A library grows from 800 books to 920 books. What is the percent increase?
Answer 15%
04 / UNDERSTAND & APPLY
Each percentage in a sequence generally applies to the current amount, so multiply the factors. A rise of 20% followed by a fall of 20% gives 1.20 × 0.80 = 0.96 of the original, a net 4% decrease. Equal percentage increases and decreases do not cancel because their bases differ.
ORIGINAL WORKED EXAMPLE
A $150 item is discounted by 20%. A 7% sales tax is then applied to the discounted price. What is the total?
Answer $128.40
05 / UNDERSTAND & APPLY
If final = factor × original, then original = final/factor. Do not “undo” a 25% discount by adding 25% of the sale price: the sale price is a smaller base. A decrease of 25% retains 75%, so divide by 0.75.
ORIGINAL WORKED EXAMPLE
A jacket costs $84 after a 30% discount. What was its original price?
Answer $120
06 / UNDERSTAND & APPLY
If A is 160% of B, then A = 1.60B. If A is 160% greater than B, then A = 2.60B because the original 100% is still present. “Of” and “greater than” determine different multipliers. A quantity that is 250% of its original value has increased by 150%.
ORIGINAL WORKED EXAMPLE
A new battery capacity is 175% of the old capacity of 48 units. What is the new capacity, and its percent increase?
Answer 84 units; 75% increase
07 / UNDERSTAND & APPLY
A rate moving from 24% to 30% rises by 6 percentage points. Its relative percent increase is (30 − 24)/24 × 100% = 25%. Percentage points subtract rates as displayed; percent change compares the difference to the old rate. Use the requested measure.
A statement such as “support doubled” means a 100% relative increase in the support measure, not necessarily an increase of 100 percentage points.
08 / UNDERSTAND & APPLY
For simple interest, interest earned is Prt and total is P(1 + rt), where P is principal, r the rate per time unit and t the matching number of time units. For annual compounding, the simplified total is P(1 + r)^t for t years. Fees, different compounding intervals and payments need separate information. These are mathematical models, not a prediction of an actual account.
For P = 500 and r = 0.04 over 3 years, simple interest gives 500(1 + 0.12) = 560. Annual compounding gives 500(1.04)³ = 562.432 before any specified rounding. The models differ because compound interest changes the base each year.
09 / UNDERSTAND & APPLY
Use the exact assumptions in each question.
ORIGINAL PRACTICE · 1
45
0.18 × 250 = 45.
ORIGINAL PRACTICE · 2
35%
42/120 = 0.35, or 35 per hundred.
ORIGINAL PRACTICE · 3
84
Multiply by 1.12: 75 × 1.12 = 84.
ORIGINAL PRACTICE · 4
$204
The retained multiplier is 0.85, giving 240 × 0.85.
ORIGINAL PRACTICE · 5
$100
Solve 1.08P = 108, so P = 100.
ORIGINAL PRACTICE · 6
1% decrease
1.10 × 0.90 = 0.99, leaving 99% of the original.
ORIGINAL PRACTICE · 7
A = 3.20B
Add the original 100% to the extra 220%, giving 320% of B.
ORIGINAL PRACTICE · 8
10 percentage points; 25% relative increase
Subtract to get 10 points; divide 10 by the old rate 40 to get 0.25.
ORIGINAL PRACTICE · 9
$90 interest
Prt = 900 × 0.05 × 2 = 90; the total including principal would be $990.
KEEP THE BIG IDEAS
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