Printable

2013 AMC 10A Problem 10

2013 AMC 10A Problem 10AMC 10 Step by Step

Name: Date: October 28, 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

Pretend there are 100 flowers: 60 pink (40 of them carnations) and 40 red (30 of them carnations). That is 70 carnations.

Solution

Each fraction talks about a different group (all flowers, pink flowers, red flowers). Using real numbers keeps them straight. The answer is a percent, so pick $100$ flowers.

Pink and red. Six tenths are pink: $60$ pink flowers. The other $40$ are red.
Pink carnations. One third of the $60$ pink flowers are roses, so $20$ pink roses. The rest, $60 - 20 = 40$ , are pink carnations.
Red carnations. Three fourths of the $40$ red flowers are carnations: $30$ red carnations (and $10$ red roses).
Put it in a table.

	Roses	Carnations	Total
Pink	20	40	60
Red	10	30	40
Total	30	70	100

Answer. $70$ carnations out of $100$ flowers is $70\%$ .

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

Why this works

A percent does not change with the size of the bouquet, so you may choose any total you like. Picking $100$ turns every fraction into a whole number of flowers. A small table with rows (color) and columns (type) makes sure each fraction is used on the right group.

The trap

Taking one third of all the flowers as roses instead of one third of the pink ones, or forgetting to turn pink roses into pink carnations.

Concise 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 $\boxed{\textbf{(E)}\ 70}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)

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