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2015 AMC 10B Problem 4

2015 AMC 10B Problem 4AMC 10 Step by Step

Name: Date: October 28, 2026

Fractions & Decimals · difficulty 1 of 5 · about 2 min · position P1-10

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?

A) $\text{Alex, Beth, Cyril, Dan}$
B) $\text{Beth, Cyril, Alex, Dan}$
C) $\text{Beth, Cyril, Dan, Alex}$
D) $\text{Beth, Dan, Cyril, Alex}$
E) $\text{Dan, Beth, Cyril, Alex}$

Answer

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.

The trap

Assuming the leftovers must be the smallest share and putting Dan last, choice (B). Dan's 13/60 is actually more than Alex's 12/60.

Concise solution

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)

Techniques

Set up the equation/formula and compute; no special trick needed