1963 TMTA High School Algebra I Contest Problems

Problem 1

If $4a - x = 3a - 4x,$ then $x =$

$\text{(A)} \quad -a \quad \text{(B)} \quad 7a/5 \quad \text{(C)} \quad a/5 \quad \text{(D)} \quad -a/3 \quad \text{(E) NOTA}$

Solution

Problem 2

The number of dimes in $K-4$ half dollars is:

$\text{(A)} \quad K-10 \quad \text{(B)} \quad 5K-20 \quad \text{(C)} \quad \frac{K-4}{5} \quad \text{(D)} \quad \frac{K}{5}-4 \quad \text{(E) NOTA}$

Solution

Problem 3

Which of the following has a negative root?

$\text{(A)} \quad 2X - 1/2 = 3$

$\text{(B)} \quad X + \frac{3}{X} = 4$

$\text{(C)} \quad \frac{3X-2}{3}= 5$

$\text{(D)} \quad 4X = 1$

$\text{(E) NOTA}$

Solution

Problem 4

The product of $3.5 \times 10^3$ and $1.8 \times 10^4$ is:

$\text{(A)} 5.3 \times 10^7 \quad \text{(B)} \quad 6.3 \times 10^7 \quad \text{(C)} \quad 6.3 \times 10^{12} \quad \text{(D)} \quad 5.3 \times 10^1 \quad \text{(E) NOTA}$ Solution

Problem 5

$\frac{\sqrt{64}}{\sqrt{25} -\sqrt{16}}$ is equal to:

$\text{(A)} \quad 8 \quad \text{(B)} \quad 8/9 \quad \text{(C)} \quad 2/5 \quad \text{(D)} \quad 8/41 \quad \text{(E)} \quad 6$ Solution

Problem 6

$(3x+2)(4x-5)$ is equal to:

$\text{(A)} \quad 12x^2-10 \quad \text{(B)} \quad 12x^2-23x-10 \quad \text{(C)} \quad 12x^3+7x-10$

$\text{(D)} \quad 12x^2-7x-10 \quad \text{(E) NOTA}$ Solution

Problem 7

$\sqrt[3]{-8}$ is equal to

$\text{(A)} \quad 2 \quad \text{(B)} \quad \pm 2 \quad \text{(C)} \quad -2 \quad \text{(D)} \quad 4 \quad \text{(E)} \quad 2\sqrt[3]{-2}$

Solution

Problem 8

$\frac{a^{-1} + b^{-1}}{(a+b)^{-1}}$ is equal to

$\text{(A)} \quad \frac{(a+b)^2}{ab} \quad \text{(B)} \quad \frac{1}{ab} \quad \text{(C)} \quad ab \quad \text{(D)} \quad \frac{ab}{(a+b)^2} \quad \text{(E)} a+b$ Solution

Problem 9

Combine: $(-3)^4 - (-3)^3$

$\text{(A)} \quad (-3)^{7} \quad \text{(B)} \quad (-3)^{1} \quad \text{(C)} \quad 3^3\cdot 2^2 \quad \text{(D)} \quad 54 \quad \text{(E)} 3^1$

Solution

Problem 10

Which of the following is true?

$\text{(A)} \quad a^3a^4 = (a^3)^7$

$\text{(B)} \quad a^3 + a^4 = a^7$

$\text{(C)} \quad \frac{(a+b)^3}{a^3} = b^3$

$\text{(D)} \quad a^3a^4 = a^12$

$\text{(E)} \quad \frac{(ab)^3}{a^3} = b^3$ Solution

Problem 11

$x^3+y^3$ factored into real primes is

$\text{(A)} \quad (x+y)(x^2+2xy+y^2) \quad \text{(B)} \quad (x+y)(x^2-2xy+y^2) \quad \text{(C)} \quad (x+y)(x^2-xy+y^2)$

$\text{(D)} \quad (x+y)(x^2+xy+y^2) \quad \text{(E)} (x+y)(x^2-y^2)$

Solution

Problem 12

Divide $bx^3$ by $\frac{x^2}{b}.$

$\text{(A)} \quad b^2x \quad \text{(B)} \quad bx^5 \quad \text{(C)} \quad \frac{x^5}{b^2} \quad \text{(D)} \quad \frac{b^2}{x} \quad \text{(E)} \quad x$

Solution

Problem 13

If a "solution set" is the set of all numbers which satisfy a statement, which is the solution set of $2x+4>16?$

$\text{(A)} \quad x>6 \quad \text{(B)} \quad x=6 \quad \text{(C)} \quad x=-6 \quad \text{(D)} \quad x<6 \quad \text{(E) NOTA}$ Solution

Problem 14

$16x^4 - y^4$ factored into its real prime factors is equal to:

$\text{(A)} \quad (4x^2+y^2)(2x-y)^2 \quad \text{(B)} \quad (4x^2+y^2)(4x^2-y)^2 \quad \text{(C)} \quad (4x^2+y^2)(2x-y)(2x+y)$

$\text{(D)} \quad 16(x^2+y^2)(x-y)(x+y) \quad \text{(E) can't be factored}$ Solution

Problem 15

$\sqrt[3]{a^5}$ is equal to

$\text{(A)} \quad a^{5/3} \quad \text{(B)} \quad a^{3/5} \quad \text{(C)} \quad \frac{1}{a} \quad \text{(D)} \quad \frac{1}{a^{3/5}} \quad \text{(E)} \quad (a^3)^{1/5}$

Solution

Problem 16

The factors of $\frac{4}{9} + \frac{1}{3}x + \frac{3}{16}x^2$ are

$\text{(A)} \quad (1/9-x/16)(4-3x) \quad \text{(B)} \quad (2/9-x/4)(2-3x/4) \quad \text{(C)} \quad (4/9-3x/16)(1-x)$

$\text{(D)} \quad -(2/3-x/4)(2/3+3x/4) \quad \text{(E)} \quad 4/9-3x(1/16+1)$

Solution

Problem 17

If the graph if the equation $5x+ky=9$ passes through the point $(3, -1,)$ what is the value of $k?$

$\text{(A)} \quad 24 \quad \text{(B)} \quad 16 \quad \text{(C)} \quad -6 \quad \text{(D)} \quad 6 \quad \text{(E)} \quad 4$ Solution

Problem 18

A dealer packed $1800$ basketballs in $6$ barrels and $6$ boxes. Later he packed $2250$ basketballs in $9$ barrels and $7$ boxes. Find the number of basketballs that a barrel will hold.

$\text{(A)} \quad 75 \quad \text{(B)} \quad 225 \quad \text{(C)} \quad 150 \quad \text{(D)} \quad 25 \quad \text{(E) NOTA}$ Solution

Problem 19

If the symbol $|x|$ is read "the absolute value of $x$ " and is equal to $x$ when $x\ge 0$ and is equal to $-x$ when $x \le 0,$ which of the following is always true?

$\text{(A)} \quad |-8|<|5| \quad \text{(B)} \quad |a+b|=|a-b| \quad \text{(C)} \quad |4-5|<5-4 \quad \text{(D)} \quad |7|=|-5-2| \quad \text{(E)} |+a|>|-a|$

Solution

Problem 20

If a piece of cloth $44$ inches long will shrink to $42$ inches when washed, to what length in inches will a $33$ inch piece of the same cloth shrink after washing?

$\text{(A)} \quad 28\frac{1}{2} \quad \text{(B)} \quad 31 \quad \text{(C)} \quad 32 \quad \text{(D)} \quad 31\frac{1}{2} \quad \text{(E)} 30$ Solution

Problem 21

On the graph chart which point has coordinates $(-5, 1)?$

$\text{(A)} \quad A \quad \text{(B)} \quad B \quad \text{(C)} \quad C \quad \text{(D)} \quad D \quad \text{(E)} \quad E$ Solution

Problem 22

On the graph chart which point has zero for its ordinate?

$\text{(A)} \quad A \quad \text{(B)} \quad B \quad \text{(C)} \quad C \quad \text{(D)} \quad D \quad \text{(E)} \quad E$ Solution

Problem 23

On the graph chart, if the lines $y+5x+24=0$ and $y+x+4=0$ were graphed, at which point would they intersect?

$\text{(A)} \quad A \quad \text{(B)} \quad B \quad \text{(C)} \quad D \quad \text{(D)} \quad E \quad \text{(E)} \quad F$

Solution

Problem 24

When $5$ is subtracted from a certain number, $N,$ $\frac{1}{5}$ of the number remains. The equation which expresses this relationship is:

$\text{(A)} \quad N-5=\frac{1}{5} \quad \quad \text{(B)} \quad \frac{1}{5}N-5=0 \quad \text{(C)} \quad 5N+25=N$

$\text{(D)} \quad N-5=\frac{1}{5}N \quad \text{(E)} \quad 5N-25=1$ Solution

Problem 25

If $\sqrt{5}=2.236,$ the value of $\frac{1+\sqrt{5}}{3-2\sqrt{5}}$ to the nearest hundredth is:

$\text{(A)} \quad -2.25 \quad \text{(B)} \quad -2.35 \quad \text{(C)} \quad -2.15 \quad \text{(D)} \quad -2.20 \quad \text{(E)} \quad -2.50$ Solution

Problem 26

If sulfuric acid is chemically pure, how many quarts of water must be added to one quart of acid to make a $10\%$ mixture?

$\text{(A)} \quad 1 \quad \text{(B)} \quad 3 \quad \text{(C)} \quad 5 \quad \text{(D)} \quad 7 \quad \text{(E)} \quad 9$

Solution

Problem 27

If $A$ varies directly as $B,$ and if $A=5$ when $B=3,$ then the value of $A$ when $B=15$ is:

$\text{(A)} \quad 25 \quad \text{(B)} \quad 50 \quad \text{(C)} \quad 75 \quad \text{(D)} \quad 3/5 \quad \text{(E)} \quad 5/3$

Solution

Problem 28

If we add its square to a certain number, the sum is $72.$ Find the number.

$\text{(A)} \quad 9 \quad \text{(B)} \quad 6 \quad \text{(C)} \quad 36 \quad \text{(D)} \quad -9 \quad \text{(E) NOTA}$ Solution

Problem 29

A mechanic agreed to work for $12$ days at $$10$ for each day that he worked, forfeiting $$5$ for each day that he was idle. At the end of the $12$ days he received $$75.$ How many days had he worked?

$\text{(A)} \quad 10 \quad \text{(B)} \quad 9 \quad \text{(C)} \quad 5 \quad \text{(D)} \quad 3 \quad \text{(E) NOTA}$

Solution

Problem 30

If $x$ is positive, which of the following is always less than one?

$\text{(A)} \quad 1/x \quad \text{(B)} \quad x^2 \quad \text{(C)} \quad \frac{x}{x+1} \quad \text{(D)} \quad \frac{1+x}{x} \quad \text{(E)} \frac{1-x}{x}$

Solution

Problem 31

Solve for $x$ when $\frac{x}{x-1} = 3 + \frac{1}{x-1}.$

$\text{(A)} \quad 2 \quad \text{(B)} \quad -1 \quad \text{(C)} \quad 1 \quad \text{(D)} \quad -2 \quad \text{(E) NOTA}$ Solution

Problem 32

A mathematical set is a collection of elements which satisfy a specified condition or conditions and braces {} are used to indicate a set. Which of the following is the set of all positive prime numbers between one and ten?

$\text{(A)} \quad \{2, 3, 4, 5, 7, 9\}$

$\text{(B)} \quad \{2, 3, 6, 9\}$

$\text{(C)} \quad \{2, 4, 6, 8\}$

$\text{(D)} \quad \{2, 3, 5, 7\}$

$\text{(E)} \quad \{3, 5, 7, 9\}$ Solution

Problem 33

From the formula $A=\frac{1}{2} H(B+C),$ the value of $B$ in terms of $A, H,$ and $C$ is:

$\text{(A)} \quad \frac{2A-HC}{H} \quad \text{(B)} \quad (2A-C)H \quad \text{(C)} \quad \frac{H}{2A-HC}$

$\text{(D)} \quad \frac{HC-2A}{H} \quad \text{(E)} \quad H-2A+C$ Solution

Problem 34

An acre of wheat yielded $2000$ pounds more of straw than of grain. The weight of the grain was $3/10$ of the total weight of grain and straw. How many $60$ pound bushels of grain were produced?

$\text{(A)} \quad 1500 \quad \text{(B)} \quad 1200 \quad \text{(C)} \quad 50 \quad \text{(D)} \quad 25 \quad \text{(E) NOTA}$

Solution

Problem 35

Combine and simplify $\frac{2}{6-3x}+\frac{5}{x-2}-\frac{3}{4-2x}$

$\text{(A)} \quad \frac{-35}{6(x-2)} \quad \text{(B)} \quad \frac{-35}{6(2-x)} \quad \text{(C)} \quad \frac{35}{6(2-x)}$

$\text{(D)} \quad \frac{6}{2-x} \quad \text{(E)} \quad \frac{6}{x-2}$ Solution

Problem 36

Find the fraction which equals 1/4 when 3 is subtracted from the numerator, but equals 1/2 when 2 is added to the denominator.

$\text{(A)} \quad 3/4 \quad \text{(B)} \quad 8/5 \quad \text{(C)} \quad 2/3 \quad \text{(D)} \quad 5/8 \quad \text{(E) NOTA}$

Solution

Problem 37

A man has $$3000$ invested at $x\%$ and $$2000$ at $(x+1)\%.$ The annual interest on the investments totals $$220.$ Find the $x$ rate.

$\text{(A)} \quad 2\% \quad \text{(B)} \quad 4\% \quad \text{(C)} \quad 3\frac{1}{2}\% \quad \text{(D)} \quad 5\% \quad \text{(E) NOTA}$ Solution

Problem 38

A farmer sold $60$ hogs for $$2030.$ Some were sold for $$ 30$ , and the remainder were sold for $$ 40$ each. How many were sold at $$ 30$ each?

$\text{(A)} \quad 37; \quad \text{(B)} \quad 19; \quad \text{(C)} \quad 29; \quad \text{(D)} \quad 42; \quad \text{(E)} \quad \text{none of these}$ Solution

Problem 39

The length of service a chair cover will give varies directly as the strength of the fabric and inversely as the amount of wear it receives. If one fabric, which is twice as strong as a second fabric and receives three times as much wear, lasts for 4 years, how long will the second fabric last?

$\text{(A)} \quad 9; \quad \text{(B)} \quad 4; \quad \text{(C)} \quad 2; \quad \text{(D)} \quad 6; \quad \text{(E)} \quad \text{none of these}$ Solution

Problem 40

If $64x^{3}-8y^{3}$ is divided by $4x-2y$ , the quotient will be:

$\text{(A)} \quad 4x^{2}-2y^{2} \quad \text{(B)} \quad 4x^{2}+2y^{2} \quad \text{(C)} \quad 16x^{2}+8xy+4y^{2}$

$\text{(D)} \quad 16x^{2}-8xy+4y^{2} \quad \text{(E)} \quad \text{none of these}$ Solution

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1963 TMTA High School Algebra I Contest Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40