1985 AJHSME Problems/Problem 6

Problem

A ream of paper containing $500$ sheets is $5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $7.5$ cm high?

$\text{(A)}\ 250 \qquad \text{(B)}\ 550 \qquad \text{(C)}\ 667 \qquad \text{(D)}\ 750 \qquad \text{(E)}\ 1250$

Solution 1

We could solve the first equation for the thickness of one sheet of paper, and divide into the 2nd equation (which is one way to do the problem), but there are other ways, too.

Let's say that $500\text{ sheets}=5\text{ cm}\Rightarrow \frac{500 \text{ sheets}}{5 \text{ cm}} = 1$ . So by multiplying $7.5 \text{ cm}$ by this fraction, we SHOULD get the number of sheets in 7.5 cm. Solving gets

$\begin{align*} \frac{7.5 \times 500}{5} &= 7.5 \times 100 \\ &= 750 \text{ sheets} \\ \end{align*}$

$750$ is $\boxed{\text{D}}$

Solution 2

We can set up a direct proportion relating the amount of sheets to the thickness because according to the problem, all the papers have the same thickness. Our proportion is $\frac{5}{500}=\frac{7.5}{x}$ where $x$ is the number we are looking for. Next, we cross-multiply to get $5x=500 \times 7.5$ so $x=750$ which is $\boxed{\text{D}}$

~GrantStar

Solution 3 (Very quick, use this to do the problem quickly)

We immediately see $500$ sheets of paper per $5$ cm is $\frac{500}{5}$ , which can be simplified to $\frac{100}{1}$ . The denominator is supposed to be $7.5$ . We can now just multiply the numerator, $100$ , by $7.5$ , to get $\boxed{\textbf{(D)}\ 750}$ . ~shunyipanda

Video Solution

https://youtu.be/Dp1i5sCWN_c

~savannahsolver

1985 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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 AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

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