1953 AMC 12

Question 1

A boy buys oranges at $3$ for $10$ cents. He will sell them at $5$ for $20$ cents. In order to make a profit of $\$1.00,$ he must sell:

$67$ oranges
$150$ oranges
$200$ oranges
an infinite number of oranges
none of these

Question 2

A refrigerator is offered for sale at $\$250.00$ less successive discounts of $20\%$ and $15\%.$ The sale price of the refrigerator is:

$35\%$ less than $\$250.00$
$65\%$ of $\$250.00$
$77\%$ of $\$250.00$
$68\%$ of $\$250.00$
none of these

Question 3

The factors of the expression $x^2+y^2$ are:

$(x+y)(x-y)$
$(x+y)^2$
$(x^{\frac{2}{3}}+y^{\frac{2}{3}})(x^{\frac{4}{3}}+y^{\frac{4}{3}})$
$(x+iy)(x-iy)$
none of these

Question 4

The roots of $x(x^2+8x+16)(4-x)=0$ are:

$0$
$0,$ $4$
$0,$ $4,$ $-4$
$0,$ $4,$ $-4,$ $-4$
none of these

Question 5

If $\log_6 x=2.5,$ the value of $x$ is:

$90$
$36$
$36\sqrt6$
$0.5$
none of these

Question 6

Charles has $5q+1$ quarters and Richard has $q+5$ quarters. The difference in their money in dimes is:

$10(q-1)$
$\dfrac25(4q-4)$
$\dfrac25(q-1)$
$\dfrac52(q-1)$
none of these

Question 7

The fraction $$
\frac{\sqrt{a^2+x^2}-\dfrac{x^2-a^2}{\sqrt{a^2+x^2}}}{a^2+x^2}
$$ reduces to:

$0$
$\dfrac{2a^2}{a^2+x^2}$
$\dfrac{2x^2}{(a^2+x^2)^{\frac{3}{2}}}$
$\dfrac{2a^2}{(a^2+x^2)^{\frac{3}{2}}}$
$\dfrac{2x^2}{a^2+x^2}$

Question 8

The value of $x$ at the intersection of $y=\dfrac8{x^2+4}$ and $x+y=2$ is:

$-2+\sqrt5$
$-2-\sqrt5$
$0$
$2$
none of these

Question 9

The number of ounces of water needed to reduce $9$ ounces of shaving lotion containing $50\%$ alcohol to a lotion containing $30\%$ alcohol is:

Question 10

The number of revolutions of a wheel, with fixed center and with an outside diameter of $6$ feet, required to cause a point on the rim to go one mile is:

$880$
$\dfrac{440}{\pi}$
$\dfrac{880}{\pi}$
$440\pi$
none of these

Question 11

A running track is the ring formed by two concentric circles. It is $10$ feet wide. The circumferences of the two circles differ by about:

$10$ feet
$30$ feet
$60$ feet
$100$ feet
none of these

Question 12

The diameters of two circles are $8$ inches and $12$ inches respectively. The ratio of the area of the smaller to the area of the larger circle is:

$\dfrac23$
$\dfrac49$
$\dfrac94$
$\dfrac12$
none of these

Question 13

A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is $18$ inches, the median of the trapezoid is:

$36$ inches
$9$ inches
$18$ inches
not obtainable from these data
none of these

Question 14

Given the larger of two circles with center $P$ and radius $p$ and the smaller with center $Q$ and radius $q.$ Draw $\overline{PQ}.$ Which of the following statements is false?

$p-q$ can be equal to $\overline{PQ}$
$p+q$ can be equal to $\overline{PQ}$
$p+q$ can be less than $\overline{PQ}$
$p-q$ can be less than $\overline{PQ}$
none of these

Question 15

A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is:

$\dfrac14$ the area of the original square
$\dfrac12$ the area of the original
square
$\dfrac12$ the area of the circular piece
$\dfrac14$ the area of the circular piece
none of these

Question 16

Adams plans a profit of $10\%$ on the selling price of an article and his expenses are $15\%$ of sales. The rate of mark-up on an article that sells for $\$5.00$ is:

$20\%$
$25\%$
$30\%$
$33\dfrac13\%$
$35\%$

Question 17

A man has part of $\$4500$ invested at $4\%$ and the rest at $6\%.$ If his annual return on each investment is the same, the average rate of interest which he realizes on the $\$4500$ is:

$5\%$
$4.8\%$
$5.2\%$
$4.6\%$
none of these

Question 18

One of the factors of $x^4+4$ is:

$x^2+2$
$x+1$
$x^2-2x+2$
$x^2-4$
none of these

Question 19

In the expression $xy^2,$ the values of $x$ and $y$ are each decreased $25\%;$ the value of the expression is:

decreased $50\%$
decreased $75\%$
decreased $\frac{37}{64}$ of its value
decreased $\frac{27}{64}$ of its value
none of these

Question 20

If $y=x+\dfrac1x,$ then $x^4+x^3-4x^2+x+1=0$ becomes:

$x^2(y^2+y-2)=0$
$x^2(y^2+y-3)=0$
$x^2(y^2+y-4)=0$
$x^2(y^2+y-6)=0$
none of these