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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 5"

(About to merge the solutions in the two pages while keeping the page clean. I will credit all contributors for the other solution. Also, reformatted Sol 2.)
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The halves are <math>\frac{15}{2},\frac{15}{6},\frac{15}{10},</math> which are <math>7\frac12,2\frac12,1\frac12,</math> respectively. We get <math>15,10,9,5,4,3</math> from this group of numbers.
 
The halves are <math>\frac{15}{2},\frac{15}{6},\frac{15}{10},</math> which are <math>7\frac12,2\frac12,1\frac12,</math> respectively. We get <math>15,10,9,5,4,3</math> from this group of numbers.
  
−
Note that <math>10</math> appears in both groups. Therefore, the answer is <math>6+6-1=\boxed{\textbf{(C)}\  11}.</math>
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The quarters are <math>\frac{15}{4},\frac{15}{12},</math> which are <math>3\frac34,1\frac14,</math> respectively. We get <math>5</math> from this group of numbers.
 +
 
 +
Note that <math>10</math> and <math>5</math> each appear twice. Therefore, the answer is <math>6+6-1=\boxed{\textbf{(C)}\  11}.</math>
  
 
~lopkiloinm
 
~lopkiloinm

Revision as of 06:46, 5 January 2022

The following problem is from both the 2021 Fall AMC 10B #7 and 2021 Fall AMC 12B #5, so both problems redirect to this page.

Problem

Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Solution 1

~KingRavi ~samrocksnature ~Wilhelm Z ~MRENTHUSIASM ~Steven Chen (www.professorchenedu.com)

Solution 2

Let $a=15-b,$ so the special fraction is \[\frac ab = \frac{15-b}{b} = \frac{15}{b}-1.\] We can ignore the $-1$ part and only focus on $\frac{15}{b}.$

The integers are $\frac{15}{1},\frac{15}{3},\frac{15}{5},$ which are $15,5,3,$ respectively. We get $30,20,18,10,8,6$ from this group of numbers.

The halves are $\frac{15}{2},\frac{15}{6},\frac{15}{10},$ which are $7\frac12,2\frac12,1\frac12,$ respectively. We get $15,10,9,5,4,3$ from this group of numbers.

The quarters are $\frac{15}{4},\frac{15}{12},$ which are $3\frac34,1\frac14,$ respectively. We get $5$ from this group of numbers.

Note that $10$ and $5$ each appear twice. Therefore, the answer is $6+6-1=\boxed{\textbf{(C)}\ 11}.$

~lopkiloinm

See Also

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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