Katherine Walls

2021-12-16

$20\le A+B\le 24$
$A+B=?$
I have A copies of $\frac{2}{3}$ on the left-hand side of the equality, and B copies of $\frac{4}{5}$ on the right. In total, I've written down between 20 and 24 fractions. Exactly how many fractions are there?

Karen Robbins

Expert

Well you have
$\frac{2}{3}A=\frac{4}{5}B⇒B=\frac{5}{6}A$
so
$A+B=\frac{11}{6}A$
Since A+B is an integer and is between 20 and 24, we must have that
$A+B=\frac{11}{6}\left(12\right)=22$

Wendy Boykin

Expert

For both fractions I made their denominator equal, so:

The smallest multiple they have on common is 60, so to get equality this means we need 6 times $\frac{2}{3}$ and 5 times $\frac{4}{5}$. Hence the answer must be a multiple of 11, so the answer is 22.

RizerMix

Expert

We have
$\frac{2}{3}A=\frac{4}{5}B;$
which gives
$B=\frac{5A}{6}$
which says that A is divided by 6.
In another hand,
$20\le \frac{5}{6}A+A\le 24$
or
$\frac{120}{11}\le A\le \frac{144}{11}$
or
$10\frac{10}{11}\le A\le 13\frac{1}{11}$
which gives
A=12,
B=10
and
A+B=22.

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