Question type

Percent of a percent on the SAT

When percents link three quantities, multiply them: if A is 25% of B and B is 60% of C, then A is $0.25 \times 0.60 = 0.15$ , or 15%, of C. "25% less than" is a factor of 0.75 and "60% greater than" is 1.60, so A is $0.75 \times 1.60 = 1.20$ times C, or 120% of it.

These questions link three quantities with two percents, such as students, juniors and juniors taking Spanish, and ask for a total, a part, or what percent one is of another.

It appeared on three of the five official practice tests we analysed, four times in all.

On 3 of 5 official practice tests
Desmos helps

Why students miss it

Applying both percents to the same whole. "25% of the juniors" is 25% of the 60%, not 25% of the class.

Adding or subtracting percents, as if $25%$ less and $60%$ more make $35%$ more.

Answering the middle quantity when the question asks about the first or the last.

How to spot it

Two percents, each of a different quantity: "60% of the students ... 25% of the juniors".
"The value of a is what percent of the value of c?"
A count for the smallest group with a question about the whole.

The method

Chain the factors

Write each relationship as a multiplication, then substitute.

" $p %$ of": factor $p /100$ .
" $p %$ more than": factor $1 + p /100$ .
" $p %$ less than": factor $1 - p /100$ .
Count given for the last link: divide by the product of the factors.

With Desmos

Desmos multiplies or divides the chain exactly.

For 12 juniors who are 25% of the 60% juniors: type 12/(0.25*0.6) and read 80.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#494Data

In a certain class, 60% of the students are juniors, and 25% of the juniors are enrolled in Spanish. If 12 juniors are enrolled in Spanish, how many students are in the class?

Pick an answer, then check it.

Show the answer and explanation

Work backwards one layer at a time. The 12 Spanish students are 25% of the juniors, so juniors

= 12 \div 0.25 = 48

. Those juniors are 60% of the class, so the class size

= 48 \div 0.60 = 80

students. Equivalently,

0.25 \times 0.60 = 0.15

of the class, and

12 \div 0.15 = 80

2Question 2 of 2#465Data

The value of a is 25% less than the value of b, and the value of b is 60% greater than the value of c. The value of a is what percent of the value of c?

Pick an answer, then check it.

Show the answer and explanation

Express everything in terms of

c

. From "

b

is 60% greater than

c

b = 1.6 c

. From "

a

is 25% less than

b

a = 0.75 b = 0.75 (1.6 c) = 1.2 c

. So

a

120%

c

. Picking

c = 100

gives

b = 160

and

a = 120

, same result.

Questions students ask

How do you find a percent of a percent?

Multiply the two percents as decimals. 25% of 60% is 0.25 × 0.60 = 0.15, which is 15%.

If a is 25% less than b and b is 60% more than c, what percent of c is a?

0.75 × 1.60 = 1.20, so a is 120% of c.

Practise this type until it is automatic

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