Non-Geometric Comparison - HSPT Quantitative

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Question

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

a)

b)

c)

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Answer

Try substituting a number in for to test this problem. Here, we try .

a) $46\cdot2=92$

b) $40\cdot2+6\cdot2=92$

c) $40\cdot2+6=86$

It is now evident that (a) and (b) are equal, but not (c)

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Question

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

a) $\frac{7}{8}$ of

b) $\frac{8}{10}$ of

c) $\frac{10}{7}$ of

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Answer

Multiply the fractions by the integers in order to compare the expressions:

a) $\frac{7\cdot10}{8}=8.75$

b) $\frac{8\cdot7}{10}=5.6$

c) $\frac{10\cdot8}{7}\approx11.4$

It is now clear that (b) is smaller than (a), which is smaller than (c).

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Question

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

a)

b) $2x\cdot2y$

c) $x\cdot x\cdot y \cdot y$

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Answer

One way to test this answer is to substitute in numbers for the variables. For example, let's say that and :

a) $2\cdot2\cdot3=12$

b) $2\cdot2\cdot2\cdot3=24$

c) $2\cdot2\cdot3\cdot3=36$

None are equal!

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Question

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

a)

b)

c)

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Answer

The property of exponents to remember here is that $(x^a)^b=x^{ab}$ .

This means that $(5^3)^2=5^{2\cdot3}=5^6$ .

.

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Question

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

a)

b)

c)

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Answer

(b) is a factored version of (a). (c), however, is not raised to the same exponent. Test this by plugging a number in for . Here, we try .

a) $6\cdot2^3=6\cdot8=48$

b) $3(2\cdot2^3)=3(2\cdot8)=3\cdot16=48$

c) $3(2\cdot2)=3\cdot4=12$

(a) and (b) are equal, but (c) is not.

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Question

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

a) percent

b)

c) $\frac{2}{100}$

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Answer

Convert each to a decimal in order to compare them:

a)

b)

c)

It is now evident that (a) is smaller than (c), which is smaller than (b).

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Question

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

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

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

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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 &

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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

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

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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) $2^{2} + 4$

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

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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) $\frac{64}{8}$

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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

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

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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

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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

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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

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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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