2015 AMC 10B Problem 4

P1-10~2 minPrint

Fractions & Decimals Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

Four siblings ordered an extra large pizza. Alex ate $\frac15$ , Beth $\frac13$ , and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?

Show solution without answering

Key insight

Write every share out of 60: Alex 12, Beth 20, Cyril 15. Dan gets the rest, 60 - 47 = 13, which is more than Alex's 12.

Solution

Fractions with different bottoms are hard to compare, so put them all over the same bottom. The numbers $5$ , $3$ and $4$ all divide $60$ .

Change each share to sixtieths:

Alex	Beth	Cyril
$\frac15 = \frac{12}{60}$	$\frac13 = \frac{20}{60}$	$\frac14 = \frac{15}{60}$

Together the first three ate $\dfrac{12 + 20 + 15}{60} = \dfrac{47}{60}$ of the pizza.
Dan got what was left: $\dfrac{60 - 47}{60} = \dfrac{13}{60}$ .
Now just compare the tops: Beth $20$ , Cyril $15$ , Dan $13$ , Alex $12$ .

From most to least: Beth, Cyril, Dan, Alex.

The answer is $\boxed{\textbf{(C)}\ \text{Beth, Cyril, Dan, Alex}}$ .

Why this works

Once every fraction has the same bottom number, comparing them is just comparing whole numbers. Finding Dan's share as "the whole pizza minus what the others ate" keeps his share on the same scale. Do not guess that leftovers are small; compute them.

Concise solution

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

Use the common denominator $60$ :
$\text{Alex } \frac15 = \frac{12}{60}, \qquad \text{Beth } \frac13 = \frac{20}{60}, \qquad \text{Cyril } \frac14 = \frac{15}{60}.$
Together they ate $\frac{12 + 20 + 15}{60} = \frac{47}{60}$ , so Dan's leftovers are $\frac{60 - 47}{60} = \frac{13}{60}$ .

Ordering the numerators $20 > 15 > 13 > 12$ gives Beth, Cyril, Dan, Alex.

The answer is $\boxed{\textbf{(C)}\ \text{Beth, Cyril, Dan, Alex}}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 11: Decimals and Fractions
Introduction to Algebra, Chapter 6: Ratios and Percents
Prealgebra, Chapter 6: Decimals (§6.3, §6.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)