Difference between revisions of "2021 Fall AMC 10B Problems/Problem 7"

Revision as of 19:04, 27 November 2021

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

The possible special fractions are: $\frac{1}{14}, \frac{2}{13}, \frac{1}{4}, \frac{4}{11}, \frac{1}{2}, \frac{2}{3}, \frac{7}{8}, \frac{8}{7}, \frac{3}{2}, 2, \frac{11}{4}, 4, \frac{13}{2}, 14$

We take the fractional parts to obtain: $\frac{1}{14}, \frac{2}{13}, \frac{1}{4}, \frac{4}{11}, \frac{1}{2}, \frac{2}{3}, \frac{7}{8}, \frac{1}{7}, \frac{1}{2}, 0, \frac{3}{4}, 0, \frac{1}{2}, 0$

Looking at any of these fractions $a,$ if there does not exist a corresponding fraction equivalent to $1-a$ in this set, then $a$ cannot possibly form an integer and we may disregard it. In this manner, we may disregard the following fractions:

$\cancel{\frac{1}{14}}, \cancel{\frac{2}{13}}, \frac{1}{4}, \cancel{\frac{4}{11}}, \frac{1}{2}, \cancel{\frac{2}{3}}, \cancel{\frac{7}{8}}, \cancel{\frac{1}{7}}, \frac{1}{2}, 0, \frac{3}{4}, 0, \frac{1}{2}, 0$

We now convert the remaining fractions into their value before taking their fractional part, and divide them into subgroups in which they can form integer with the other fractions in that group. $(1)~~\frac{1}{4}, \frac{11}{4} \qquad (2)~~\frac{1}{2}, \frac{3}{2}, \frac{13}{2} \qquad (3)~~ 2,4,14$

Now, we simply try all combinations of two fractions within each subgroup to obtain

$(1)$ $\frac{1}{4}+\frac{11}{4}=3$ $(2)$ $\frac{1}{2}+\frac{1}{2}=1$ $\frac{1}{2}+\frac{3}{2}=2$ $\frac{1}{2}+\frac{13}{2}=7$ $\frac{3}{2}+\frac{3}{2}+3$ $\frac{3}{2}+\frac{13}{2}=8$ $\frac{13}{2}+\frac{13}{2}=13$ $(3)$ $2+2=4$ $2+4=6$ $2+14=16$ $4+4=8$ $4+14=18$ $14+14=28.$

Our final list of integer that may be formed consists of $1,2,3,4,6,7,8,13,16,18,28 \implies \boxed{\textbf{(C)}\ 11}.$

~samrocksnature

Solution 2

All special fractions are: $\frac{1}{14}$ , $\frac{2}{13}$ , $\frac{3}{12}$ , $\frac{4}{11}$ , $\frac{5}{10}$ , $\frac{6}{9}$ , $\frac{7}{8}$ , $\frac{8}{7}$ , $\frac{9}{6}$ , $\frac{10}{5}$ , $\frac{11}{4}$ , $\frac{12}{3}$ , $\frac{13}{2}$ , $\frac{14}{1}$ .

Hence, the following numbers are integers: $\frac{3}{12} + \frac{11}{4}$ , $\frac{5}{10} + \frac{5}{10}$ , $\frac{5}{10} + \frac{9}{6}$ , $\frac{5}{10} + \frac{13}{2}$ , $\frac{9}{6} + \frac{9}{6}$ , $\frac{9}{6} + \frac{13}{2}$ , $\frac{10}{5} + \frac{10}{5}$ , $\frac{10}{5} + \frac{12}{3}$ , $\frac{10}{5} + \frac{14}{1}$ , $\frac{12}{3} + \frac{12}{3}$ , $\frac{12}{3} + \frac{14}{1}$ , $\frac{13}{2} + \frac{13}{2}$ , $\frac{14}{1} + \frac{14}{1}$ .

This leads to the following distinct integers: 3, 1, 2, 7, 8, 4, 6, 16, 18, 13, 28.

Therefore, the answer is $\boxed{\textbf{(C) }11}$ .

~Steven Chen (www.professorchenedu.com)

Video Solution by Interstigation

https://youtu.be/p9_RH4s-kBA?t=810

@@ Line 21: / Line 21: @@
 Now, we simply try all combinations of two fractions within each subgroup to obtain
 <math>(1)</math> <cmath>\frac{1}{4}+\frac{11}{4}=3</cmath>
 <math>(2)</math> <cmath>\frac{1}{2}+\frac{1}{2}=1</cmath> <cmath>\frac{1}{2}+\frac{3}{2}=2</cmath> <cmath>\frac{1}{2}+\frac{13}{2}=7</cmath> <cmath>\frac{3}{2}+\frac{3}{2}+3</cmath> <cmath>\frac{3}{2}+\frac{13}{2}=8</cmath> <cmath>\frac{13}{2}+\frac{13}{2}=13</cmath>

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