2013 AMC 10A Problem 10

P1-10~2 minPrint

Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed Test small/specific values or special cases to find or verify the answer

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?

Show solution without answering

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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)