How to make non-geometric comparisons - HSPT Quantitative

Card 0 of 20

Question

$\dpi{100} \frac{1}{3}$ of what number is equal to 2 times 4?

Answer

Set up the following equation.

$\dpi{100} \frac{1}{3}x=2\cdot 4$

$\dpi{100} \frac{1}{3}x=8$

$\dpi{100} 3\cdot \frac{1}{3}x=8\cdot 3$

$\dpi{100} x=24$

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Question

Examine (A), (B), and (C) and find the best answer.

(A) $\dpi{100} .75$

(B) $\dpi{100} \frac{1}{2}$ of $\dpi{100} 1.5$

(C) $\dpi{100} \frac{3}{4}$

Answer

All of these choices are equal.

$\dpi{100} \frac{3}{4}$ is .75 in fraction form, and .75 is $\dpi{100} \frac{3}{4}$ in decimal form.

$\dpi{100} \frac{1}{2}$ of 1.5 is the same as $\dpi{100} \frac{1}{2}\cdot \frac{3}{2}=\frac{3}{4}$ .

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Question

Examine (A), (B), and (C) and find the best answer if both and are less than zero.

(A) $\dpi{100} -2x$

(B) $\dpi{100} -2(x+y)$

(C) $\dpi{100} -3(x+y)$

Answer

This is a difficult problem. Since $\dpi{100} x$ and $\dpi{100} y$ are both negative, then $\dpi{100} x+y$ must be less than $\dpi{100} x$ .

In (A), (B), and (C) the variables (which are negative) are all multiplied by a negative number, so the ultimate values for each is positive.

Thus, since this is $\dpi{100} negative \times negative=positive$ , the larger the absolute value of the variables AND the coefficient, the larger the answer will be.

$\dpi{100} x+y$ has an absolute value that is greater than $\dpi{100} x$ .

$\dpi{100} -3$ has an absolute value that is greater than $\dpi{100} -2$ .

Combine these two and we realize that $\dpi{100} -3(x+y)$ must be the greatest of the three choices.

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Question

Examine (a), (b), and (c) and find the best answer.

a) the square root of

b) $\frac{1}{20}$ of

c) the average of &

Answer

a) The square root of is , because $11\cdot 11=121$ .

b) $\frac{1}{20}$ of is , because $200\div 20=10$ .

c) The average of and is , because $\frac{5+15}{2}=10$ .

Therefore (b) and (c) are equal, and they are both smaller than (a).

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Question

Examine (a), (b), and (c) and find the best answer.

a) percent of

b) percent of

c) percent of

Answer

a) percent of is because $96\cdot .50=48$ .

b) percent of is because $120\cdot .30=40$ .

c) percent of is because $111\cdot .60=66$ .

Therefore (b) is smaller than (a) which is smaller than (c).

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Question

Examine (a), (b), and (c) and find the best answer.

a) $2^{(2+3)}$

b)

c) $\frac{2}{(2+3)}$

Answer

a) $2^{(2+3)}$ $= 2^{5} = 2\cdot 2\cdot 2\cdot 2\cdot 2 = 32$

b) $= 2\cdot 5 = 10$

c) $\frac{2}{(2+3)}$ $=\frac{2}{5}= .4$

Therefore (a) is larger than (b) which is larger than (c).

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Question

Examine (a), (b), and (c) and find the best answer.

a) $4^{2}$

b) $2^{4}$

c) $2^{2} + 4$

Answer

a) $4^{2} = 4\cdot 4 =16$

b) $2^{4} = 2\cdot 2\cdot 2\cdot 2 = 16$

c) $2^{2} + 4 = 2\cdot 2 + 4 = 4+4 = 8$

Therefore (a) and (b) are equal, and they are larger than (c).

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Question

Examine (a), (b), and (c) and find the best answer.

a) $5+\frac{6+8}{2}$

b) $\frac{5(6+8)}{2}$

c) $\frac{5}{2}\cdot6+8$

Answer

This question tests your understanding of the order of operations. First complete operations in parentheses, then multiplication and division, and finally addition and subtraction.

a) $5+\frac{6+8}{2} = 5+\frac{14}{2} = 5+7 = 12$

b) $\frac{5(6+8)}{2}=\frac{5\cdot 14}{2}=\frac{70}{2}=35$

c) $\frac{5}{2}\cdot6+8=2.5\cdot6+8=15+8=23$

Therefore (a) is smaller than (c) which is smaller than (b).

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Question

Examine (a), (b), and (c) and find the best answer.

a) $\sqrt{64}$

b) percent of

c) $\frac{64}{8}$

Answer

a) $\sqrt{64}$

b) percent of $= 64\cdot.10=6.4$

c) $\frac{64}{8}$ $=64\div8=8$

Therefore (a) and (c) are the same, and they are both larger than (b).

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Question

Examine (a), (b), and (c) and choose the best answer.

a) percent of percent of

b) percent of percent of

c) percent of percent of

Answer

a) percent of percent of

$.85\cdot.15\cdot400=51$

b) percent of percent of

$4.00\cdot.85\cdot15=51$

c) percent of percent of

$.15\cdot4.00\cdot85=51$

Therefore (a), (b), and (c) are all equal.

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Question

Examine (a), (b), and (c) to find the best answer:

a) $\frac{3}{x^{2}}$

b) $\left(\frac{3}{x}\right)^{2}$

c) $\frac{3^{2}}{x^{2}}$

Answer

a) $\frac{3}{x^{2}}$

This expression is already simplified.

b) $\left(\frac{3}{x}\right)^{2}$

This expression simplifies to $\left(\frac{3}{x}\right)^{2}=\frac{3^{2}}{x^{2}}=\frac{9}{x^{2}}$ .

c) $\frac{3^{2}}{x^{2}}$

This expression also simplifies to $\frac{3^{2}}{x^{2}}=\frac{9}{x^{2}}$ .

Clearly (b) and (c) are equal, but (a) is smaller because it has a smaller numerator.

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Question

Examine (a), (b), and (c) to find the best answer:

a) $11^{2}$

b) The smallest prime number larger than

c) percent of

Answer

a) $11^{2}$ $= 11\cdot11=121$

b) The smallest prime number larger than is .

c) percent of

$120\cdot.95=114$

Therefore (b) is smaller than (c) which is smaller than (a).

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Question

Examine (a), (b), and (c) to find the best answer:

a)

b)

c)

* is a non-zero integer

Answer

a) $= 4^2 \cdot a^2 = 16a^2$

b)

c)

Therefore (a) and (b) are equal. For all non-zero integers (whole numbers other than zero), will be smaller than , so (c) is smaller than (a) and (b).

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Question

Examine (a), (b), and (c) to find the best answer:

a) percent of

b) percent of

c) percent of

Answer

In each of these scenarios, if the percentage increases, the number decreases by the same factor. All cases are the same value:

a) percent of

$500\cdot.42=210$

b) percent of

$1000\cdot.21=210$

c) percent of

$250\cdot.84=210$

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Question

Examine (a), (b), and (c) to find the best answer:

a) $(2+7)\cdot8$

b) $2+(7\cdot8)$

c) $(2\cdot8)+7$

Answer

Always do the operations in parantheses first, then multiplication, then addition.

a) $(2+7)\cdot8 = 9\cdot8 = 72$

b) $2+(7\cdot8) = 2+56 =58$

c) $(2\cdot8)+7 = 16+7 = 23$

Therefore (a) is greater than (b), which is greater than (c) .

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Question

Examine (a), (b), and (c) to find the best answer:

a) $\frac{2}{3}$ of

b) $\frac{5}{2}$ of

c) $\frac{5}{6}$ of

Answer

Multiply each fraction by the number to find each value:

a) $\frac{2}{3}$ of

$\frac{36\cdot2}{3}=24$

b) $\frac{5}{2}$ of

$\frac{36\cdot5}{2}=90$

c) $\frac{5}{6}$ of

$\frac{72\cdot5}{6}=60$

Therefore (a) is less than (c), which is less than (b).

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Question

Examine (a), (b), and (c) to find the best answer:

a)

b)

c)

Answer

Simplify each expression to see if they are equal:

a) (already simplified)

b)

c)

Therefore (a) and (c) are equal, but (b) is different.

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Question

Examine (a), (b), and (c) to find the best answer:

a) $\frac{2}{3}$

b) percent

c)

Answer

Convert each expression into a decimal in order to compare them:

a) $.\bar{66}$

b)

c)

Therefore (a) is the largest.

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Question

Examine (a), (b), and (c) to find the best answer:

a) $\frac{1}{2}$ of

b) $\frac{1}{3}$ of

c) $\frac{3}{4}$ of

Answer

Calculate each expression in order to compare them:

a) $\frac{1}{2}$ of

$\frac{32}{2}=16$

b) $\frac{1}{3}$ of

$\frac{45}{3}=15$

c) $\frac{3}{4}$ of

$\frac{3\cdot20}{4}=15$

(b) and (c) are equal, and (a) is greater than both.

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Question

Examine (a), (b), and (c) to find the best answer:

a) $\frac{1}{5}$

b) $\frac{3}{10}$

c) $\frac{1}{10}$

Answer

Rewrite the first fraction with a denominator of in order to compare more easily:

a) $\frac{1}{5}=\frac{2}{10}$

b) $\frac{3}{10}$

c) $\frac{1}{10}$

It becomes clear that (b) is the greatest, followed by (a), then (c).

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