AMC 10 · 2009 · #5

Grade 7 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

percentagefraction-arithmeticlinear-equations-one-var work-backwards ↑ Prerequisites: percentage

📏 Short solution 💡 2 insights

Problem

Twenty percent less than 60 is one-third more than what number?

Pick an answer.

(A)

$\ 16$

(B)

$\ 30$

(C)

$\ 32$

(D)

$\ 36$

(E)

$\ 48$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: First read what 'twenty percent less than 60' equals. That result is described as being one-third more than some hidden number. Find the hidden number.

Givens: The starting quantity is 60.; 'Twenty percent less than 60' is a single value you can compute.; That value equals 'one-third more than' an unknown number.

Unknowns: The unknown number n whose 'one-third more' equals twenty percent less than 60.

Understand

Restated: First read what 'twenty percent less than 60' equals. That result is described as being one-third more than some hidden number. Find the hidden number.

Givens: The starting quantity is 60.; 'Twenty percent less than 60' is a single value you can compute.; That value equals 'one-third more than' an unknown number.

Plan

Primary tool: #11 Work Backwards

Secondary: #7 Identify Subproblems, #4 Introduce a Variable

The sentence hands you the finished value on the left and asks for the starting number on the right, so the natural move is to compute the left side, then undo the 'one-third more' step to walk back to the original number.

Execute — Answer: D

#7 Identify Subproblems 7.RP.A.3 Step 1

Compute twenty percent less than 60

Split the sentence into two halves.
The left half is a plain calculation: twenty percent of 60 is 12, and 'less than' means you subtract it from 60.

$$60 - 0.20 \times 60 = 60 - 12 = 48$$

💡 'Percent less' means slice that share off the whole, not add it on.

#4 Introduce a Variable 7.RP.A.3 Step 2

Write 'one-third more than n'

Call the hidden number n.
'One-third more than n' means you start with n and add another third of n, which combines into four-thirds of n.
That must equal the 48 from the left half.

$$n + \frac{1}{3}n = \frac{4}{3}n = 48$$

💡 'One-third more' scales the number up to 4/3 of itself, not just to a third of it.

#11 Work Backwards 6.EE.B.7 Step 3

Undo the scaling to find n

You have four-thirds of n equal to 48, so reverse the multiplication: divide 48 by 4/3, which is the same as multiplying 48 by 3/4.
That gives n = 36, which is choice (D).

$$n = 48 \div \frac{4}{3} = 48 \times \frac{3}{4} = 36$$

💡 To find the original from a scaled-up value, divide by the same factor you multiplied by.

[1] #7 7.RP.A.3 Split the sentence into two halves. The left half is a plain calculation: twenty

[2] #4 7.RP.A.3 Call the hidden number n. 'One-third more than n' means you start with n and add

[3] #11 6.EE.B.7 You have four-thirds of n equal to 48, so reverse the multiplication: divide 48

Review

Reasonableness: Check the answer forward: one-third more than 36 is 36 + 12 = 48, and twenty percent less than 60 is also 48. Both sides match, so 36 is right. A common trap is taking one-third of 48 to get 16 (choice A); that misreads 'more than' as 'of'.

Alternative: Set up one equation directly: (4/3)n = (4/5)(60). Simplify the right side to 48 and solve for n = 48 * 3/4 = 36. Same answer with the algebra done in one line.

CCSS standards used (min grade 7)

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems (Reading 'twenty percent less than 60' as 48 and 'one-third more than n' as four-thirds of n.)
6.EE.B.7 Solve real-world problems by writing and solving equations of the form px = q (Solving (4/3)n = 48 for the unknown number n.)

⭐ 'One-third more' means times four-thirds, so to go back you divide by four-thirds.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Compute twenty percent less than 60

Write 'one-third more than n'

Undo the scaling to find n

Review

CCSS standards used (min grade 7)

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