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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 $(-5, 1)?$

1963 Algebra I 22.PNG

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

Solution

Problem 22

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

1963 Algebra I 22.PNG

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

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?

1963 Algebra I 22.PNG

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

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

See Also

1963 TMTA High School Algebra I Contest (Problems)
Preceded by
First Contest 		TMTA High School Mathematics Contest Followed by
1964 TMTA High School Mathematics Contests
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