1951 AHSME Problems

Problem 1

The percent that $M$ is greater than $N$ is:

$\mathrm{(A) \ } \frac {100(M - N)}{M} \qquad \mathrm{(B) \ } \frac {100(M - N)}{N} \qquad \mathrm{(C) \ } \frac {M - N}{N} \qquad \mathrm{(D) \ } \frac {M - N}{M} \qquad \mathrm{(E) \ } \frac {100(M + N)}{N}$

Solution

Problem 2

A rectangular field is half as wide as it is long and is completely enclosed by $x$ yards of fencing. The area in terms of $x$ is:

$(\mathrm{A})\ \frac{x^2}2 \qquad (\mathrm{B})\ 2x^2 \qquad (\mathrm{C})\ \frac{2x^2}9 \qquad (\mathrm{D})\ \frac{x^2}{18} \qquad (\mathrm{E})\ \frac{x^2}{72}$

Solution

Problem 3

If the length of a diagonal of a square is $a + b$ , then the area of the square is:

$\mathrm{(A) \ (a+b)^2 } \qquad \mathrm{(B) \ \frac{1}{2}(a+b)^2 } \qquad \mathrm{(C) \ a^2+b^2 } \qquad \mathrm{(D) \ \frac {1}{2}(a^2+b^2) } \qquad \mathrm{(E) \ \text{none of these} }$

Solution

Problem 4

A barn with a flat roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:

$\mathrm{(A) \ } 360 \qquad \mathrm{(B) \ } 460 \qquad \mathrm{(C) \ } 490 \qquad \mathrm{(D) \ } 590 \qquad \mathrm{(E) \ } 720$

Solution

Problem 5

Mr. A owns a home worth $10,000$ dollars. He sells it to Mr. B at a $10 \%$ profit based on the worth of the house. Mr. B sells the house back to Mr. A at a $10 \%$ loss. Then:

$\mathrm{(A) \ } \text{A comes out even.} \qquad\mathrm{(B) \ } \text{A makes } $$ \text{1100 on the deal} \qquad\mathrm{(C) \ } \text{A makes } $ $\text{1000 on the deal} \qquad \mathrm{(D) \ } \text{A loses } $$ \text{900 on the deal} \qquad\mathrm{(E) \ } \text{A loses } $ $\text{1000 on the deal}$

Solution

Problem 6

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to:

$\mathrm{(A) \ } \text{the volume of the box} \qquad \mathrm{(B) \ } \text{the square root } $$ \text{of the volume} \qquad \mathrm{(C) \ } \text{twice the volume} \qquad \mathrm{(D) \ } \text{the square of the volume} \qquad\mathrm{(E) \ } \text{the cube of the volume} $[[1951 AHSME Problems/Problem 6|Solution]]

== Problem 7 ==

An error of$ (Error compiling LaTeX. Unknown error_msg).02" $is made in the measurement of a line$ 10" $long, while an error of only$ .2" $is made in a measurement of a line$ 100" $long. In comparison with the relative error of the first measurement, the relative error of the second measurement is:$ \mathrm{(A) \ } \text{greater by }.18 \qquad\mathrm{(B) \ } \text{the same} \qquad \mathrm{(C) \ } \text{less} \qquad\mathrm{(D) \ } 10\text{ times as great} \qquad\mathrm{(E) \ } \text{correctly described by both} $[[1951 AHSME Problems/Problem 7|Solution]]

== Problem 8 ==

The price of an article is cut$ (Error compiling LaTeX. Unknown error_msg)10 \%. $To restore it to its former value, the new price must be increased by:$ \mathrm{(A) \ } 10 \% \qquad\mathrm{(B) \ } 9 \% \qquad \mathrm{(C) \ } 11\frac{1}{9} \% \qquad\mathrm{(D) \ } 11 \% \qquad\mathrm{(E) \ } \text{none of these answers} $[[1951 AHSME Problems/Problem 8|Solution]]

== Problem 9 ==

An equilateral triangle is drawn with a side length of$ (Error compiling LaTeX. Unknown error_msg)a. $A new equilateral triangle is formed by joining the midpoints of the sides of the first one. then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. the limit of the sum of the perimeters of all the triangles thus drawn is:$ \mathrm{(A) \ } \text{Infinite} \qquad\mathrm{(B) \ } 5\frac{1}{4}a \qquad \mathrm{(C) \ } 2a \qquad\mathrm{(D) \ } 6a \qquad\mathrm{(E) \ } 4\frac{1}{2}a $[[1951 AHSME Problems/Problem 9|Solution]]

== Problem 10 ==

[[1951 AHSME Problems/Problem 10|Solution]]

== Problem 11 ==

[[1951 AHSME Problems/Problem 11|Solution]]

== Problem 12 ==

[[1951 AHSME Problems/Problem 12|Solution]]

== Problem 13 ==

[[1951 AHSME Problems/Problem 13|Solution]]

== Problem 14 ==

[[1951 AHSME Problems/Problem 14|Solution]]

== Problem 15 ==

[[1951 AHSME Problems/Problem 15|Solution]]

== Problem 16 == If in applying the quadratic formula to a quadratic equation

<cmath>f(x) \equiv ax^2 + bx + c = 0,</cmath>

it happens that$ (Error compiling LaTeX. Unknown error_msg)c = \frac{b^2}{4a} $, then the graph of$ y = f(x) $will certainly:$ \mathrm{(A) \ have\ a\ maximum } \qquad \mathrm{(B) \ have\ a\ minimum} \qquad$$ (Error compiling LaTeX. Unknown error_msg)\mathrm{(C) \ be\ tangent\ to\ the\ xaxis} \qquad$$ (Error compiling LaTeX. Unknown error_msg)\mathrm{(D) \ be\ tangent\ to\ the\ yaxis} \qquad$$ (Error compiling LaTeX. Unknown error_msg)\mathrm{(E) \ lie\ in\ one\ quadrant\ only}$