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2013 AMC 10A Problem 10

2013 AMC 10A Problem 10AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P1-10

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

A)
$15$
B)
$30$
C)
$40$
D)
$60$
E)
$70$

Answer

Key insight

Take 100 flowers: 60 pink of which 40 are carnations, 40 red of which 30 are carnations, so 70 carnations.

Solution

Percentages do not depend on the size of the bouquet, so suppose there are $100$ flowers.

Six tenths are pink: $60$ pink flowers and $40$ red flowers.

One third of the pink flowers are roses, so $20$ are pink roses and the other $60 - 20 = 40$ are pink carnations.

Three fourths of the red flowers are carnations: $\tfrac34 \cdot 40 = 30$ red carnations.

Carnations in total: $40 + 30 = 70$ out of $100$ , which is $70\%$ .

The answer is $\boxed{\textbf{(E)}\ 70}$ .

Why this works

Each fraction in the problem refers to a different base (all flowers, pink flowers, red flowers). Choosing a convenient total makes every base a concrete number, so no fraction is ever applied to the wrong group. A two-by-two table (color by type) organizes the four counts and shows the row and column sums at a glance.

The trap

Applying 'one third are roses' to all the flowers instead of only the pink ones, or forgetting to convert pink roses into pink carnations.

Common mistakes

Applying 'one third are roses' to all the flowers instead of only the pink ones, or forgetting to convert pink roses into pink carnations.
Reporting the pink carnations alone ( $40\%$ ) and stopping before adding the red carnations.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer