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HSPT Math : Arithmetic

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Example Questions

Example Question #61 : Fractions

Evaluate:

$20 - 7.5 \times\left ( 3 - 1.2 \right )$ $\displaystyle 20 - 7.5 \times\left ( 3 - 1.2 \right )$

Possible Answers:

36.3 $\displaystyle 36.3$

6.5 $\displaystyle 6.5$

- 3.7 $\displaystyle - 3.7$

22.5 $\displaystyle 22.5$

-1.3 $\displaystyle -1.3$

Correct answer:

6.5 $\displaystyle 6.5$

Explanation:

By the order of operations, carry out the operation in parentheses, which is the rightmost subtraction, then the multiplication, then the leftmost subtraction:

$20 - 7.5 \times\left ( 3 - 1.2 \right )$ $\displaystyle 20 - 7.5 \times\left ( 3 - 1.2 \right )$

$= 20 - 7.5 \times 1.8$ $\displaystyle = 20 - 7.5 \times 1.8$

$\displaystyle = 20 - 13.5$

= 6.5 $\displaystyle = 6.5$

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Example Question #56 : Identities And Properties

Which of the following statements demonstrates the identity property of multiplication?

Possible Answers:

None of the examples in the other responses demonstrates the identity property of multiplication.

$\frac{2}{3} \times 1 = \frac{2}{3}$ $\displaystyle \frac{2}{3} \times 1 = \frac{2}{3}$

$\left (\frac{2}{3} \times \frac{5}{7} \right ) \times \frac{6}{11} = \frac{2}{3} \times\left ( \frac{5}{7} \times \frac{6}{11} \right )$ $\displaystyle \left (\frac{2}{3} \times \frac{5}{7} \right ) \times \frac{6}{11} = \frac{2}{3} \times\left ( \frac{5}{7} \times \frac{6}{11} \right )$

$\frac{2}{3} \times \frac{5}{7} = \frac{5}{7} \times \frac{2}{3}$ $\displaystyle \frac{2}{3} \times \frac{5}{7} = \frac{5}{7} \times \frac{2}{3}$

$\frac{2}{3} \times \frac{3}{2} = 1$ $\displaystyle \frac{2}{3} \times \frac{3}{2} = 1$

Correct answer:

$\frac{2}{3} \times 1 = \frac{2}{3}$ $\displaystyle \frac{2}{3} \times 1 = \frac{2}{3}$

Explanation:

The identity property of multiplication states that there is a number 1, called the multiplicative identity, that can be multiplied by any number to obtain that number. Of the four statements,

$\frac{2}{3} \times 1 = \frac{2}{3}$ $\displaystyle \frac{2}{3} \times 1 = \frac{2}{3}$

demonstrates this property.

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Example Question #1 : Fractions

$\displaystyle Z$ is larger than 0. Which of the following could be equal to $3 \times Z$ $\displaystyle 3 \times Z$?

I. $\displaystyle 6$

II. $2\frac{2}{7}$ $\displaystyle 2\frac{2}{7}$

III. $\displaystyle 8$

Possible Answers:

I and II

I only

II only

III only

I, II and III

Correct answer:

I, II and III

Explanation:

All of the answers can be divided by 3 to yield an answer larger than zero. In fact, any positive number would be a viable answer.

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Example Question #1 : How To Divide Fractions

Express the quotient as a fraction in lowest terms:

$5 \frac{3}{5} \div 4 \frac{1}{5}$ $\displaystyle 5 \frac{3}{5} \div 4 \frac{1}{5}$

Possible Answers:

$1 \frac{1}{4}$ $\displaystyle 1 \frac{1}{4}$

$1 \frac{1}{3}$ $\displaystyle 1 \frac{1}{3}$

$\frac{3}{4}$ $\displaystyle \frac{3}{4}$

$\frac{4}{5}$ $\displaystyle \frac{4}{5}$

$1 \frac{3}{4}$ $\displaystyle 1 \frac{3}{4}$

Correct answer:

$1 \frac{1}{3}$ $\displaystyle 1 \frac{1}{3}$

Explanation:

Rewrite the mixed fractions as improper fractions, change to a multiplication by inverting the second, cross-cancel, and multiply across:

$5 \frac{3}{5} \div 4 \frac{1}{5} = \frac{28}{5 } \div \frac{21}{5 } = \frac{28}{5 } \cdot \frac{5}{21 }= \frac{4}{1 } \cdot \frac{1}{3 }= \frac{4}{3 }$ $\displaystyle 5 \frac{3}{5} \div 4 \frac{1}{5} = \frac{28}{5 } \div \frac{21}{5 } = \frac{28}{5 } \cdot \frac{5}{21 }= \frac{4}{1 } \cdot \frac{1}{3 }= \frac{4}{3 }$

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Example Question #351 : Arithmetic

Express the quotient as a fraction in lowest terms:

$4 \frac{1}{5} \div 5 \frac{3}{5}$ $\displaystyle 4 \frac{1}{5} \div 5 \frac{3}{5}$

Possible Answers:

$\frac{5}{7}$ $\displaystyle \frac{5}{7}$

$1 \frac{1}{4}$ $\displaystyle 1 \frac{1}{4}$

$1 \frac{1}{3}$ $\displaystyle 1 \frac{1}{3}$

$\frac{6}{7}$ $\displaystyle \frac{6}{7}$

$\frac{3}{4}$ $\displaystyle \frac{3}{4}$

Correct answer:

$\frac{3}{4}$ $\displaystyle \frac{3}{4}$

Explanation:

Rewrite the mixed fractions as improper fractions, change to a multiplication by inverting the second, cross-cancel, and multiply across:

$4 \frac{1}{5} \div 5 \frac{3}{5} = \frac{21}{5} \div \frac{28}{5} = \frac{21}{5} \cdot \frac{5}{28} = \frac{3}{1} \cdot \frac{1}{4} = \frac{3}{4}$ $\displaystyle 4 \frac{1}{5} \div 5 \frac{3}{5} = \frac{21}{5} \div \frac{28}{5} = \frac{21}{5} \cdot \frac{5}{28} = \frac{3}{1} \cdot \frac{1}{4} = \frac{3}{4}$

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Example Question #352 : Arithmetic

Evaluate:

$\frac{12}{5} \div \frac{4}{25}$ $\displaystyle \frac{12}{5} \div \frac{4}{25}$

Possible Answers:

$\displaystyle 15$

$\frac{8}{25}$ $\displaystyle \frac{8}{25}$

$\frac{3}{5}$ $\displaystyle \frac{3}{5}$

$\frac{1}{15}$ $\displaystyle \frac{1}{15}$

$\frac{5}{3}$ $\displaystyle \frac{5}{3}$

Correct answer:

$\displaystyle 15$

Explanation:

Multiply by the reciprocal, cross-cancel, then multiply numerators and denominators:

$\frac{12}{5} \div \frac{4}{25} = \frac{12}{5} \times \frac{25}{4} = \frac{3}{1} \times \frac{5}{1} = \frac{15}{1} = 15$ $\displaystyle \frac{12}{5} \div \frac{4}{25} = \frac{12}{5} \times \frac{25}{4} = \frac{3}{1} \times \frac{5}{1} = \frac{15}{1} = 15$

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Example Question #353 : Arithmetic

Evaluate:

$\frac{4}{25} \div \frac{12}{5}$ $\displaystyle \frac{4}{25} \div \frac{12}{5}$

Possible Answers:

$\frac{5}{3}$ $\displaystyle \frac{5}{3}$

$\frac{1}{15}$ $\displaystyle \frac{1}{15}$

$\frac{3}{5}$ $\displaystyle \frac{3}{5}$

$\displaystyle 15$

$\frac{8}{25}$ $\displaystyle \frac{8}{25}$

Correct answer:

$\frac{1}{15}$ $\displaystyle \frac{1}{15}$

Explanation:

Multiply by the reciprocal, cross-cancel, then multiply numerators and denominators:

$\frac{4}{25} \div \frac{12}{5} = \frac{4}{25} \times \frac{5}{12} = \frac{1}{5} \times \frac{1}{3} = \frac{1}{15}$ $\displaystyle \frac{4}{25} \div \frac{12}{5} = \frac{4}{25} \times \frac{5}{12} = \frac{1}{5} \times \frac{1}{3} = \frac{1}{15}$

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Example Question #354 : Arithmetic

Evaluate:

$\frac{5}{8} \div \frac{25}{16}$ $\displaystyle \frac{5}{8} \div \frac{25}{16}$

Possible Answers:

$\frac{2}{5}$ $\displaystyle \frac{2}{5}$

$\frac{5}{2}$ $\displaystyle \frac{5}{2}$

$\frac{1}{10}$ $\displaystyle \frac{1}{10}$

$\frac{5}{4}$ $\displaystyle \frac{5}{4}$

$\frac{4}{5}$ $\displaystyle \frac{4}{5}$

Correct answer:

$\frac{2}{5}$ $\displaystyle \frac{2}{5}$

Explanation:

Multiply by the reciprocal, cross-cancel, then multiply numerators and denominators:

$\frac{5}{8} \div \frac{25}{16} = \frac{5}{8} \times \frac{16}{25} = \frac{1}{1} \times \frac{2}{5} = \frac{2}{5}$ $\displaystyle \frac{5}{8} \div \frac{25}{16} = \frac{5}{8} \times \frac{16}{25} = \frac{1}{1} \times \frac{2}{5} = \frac{2}{5}$

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Example Question #4 : Fractions

Evaluate:

$\frac{8}{5} \div \frac{24}{35}$ $\displaystyle \frac{8}{5} \div \frac{24}{35}$

Possible Answers:

$\frac{8}{3}$ $\displaystyle \frac{8}{3}$

$\frac{1}{21}$ $\displaystyle \frac{1}{21}$

$\frac{3}{8}$ $\displaystyle \frac{3}{8}$

$\frac{7}{3}$ $\displaystyle \frac{7}{3}$

$\frac{3}{7}$ $\displaystyle \frac{3}{7}$

Correct answer:

$\frac{7}{3}$ $\displaystyle \frac{7}{3}$

Explanation:

Multiply by the reciprocal, cross-cancel, then multiply numerators and denominators:

$\frac{8}{5} \div \frac{24}{35} = \frac{8}{5} \times \frac{35}{24} = \frac{1}{1} \times \frac{7}{3} =\frac{7}{3}$ $\displaystyle \frac{8}{5} \div \frac{24}{35} = \frac{8}{5} \times \frac{35}{24} = \frac{1}{1} \times \frac{7}{3} =\frac{7}{3}$

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Example Question #355 : Arithmetic

Evaluate:

$3 \div \frac{1}{12}$ $\displaystyle 3 \div \frac{1}{12}$

Possible Answers:

$\displaystyle 4$

$\displaystyle 36$

$\frac{1}{36}$ $\displaystyle \frac{1}{36}$

$\displaystyle 48$

$\frac{1}{4}$ $\displaystyle \frac{1}{4}$

Correct answer:

$\displaystyle 36$

Explanation:

$3 \div \frac{1}{12} = \frac{3}{1} \div \frac{1}{12} = \frac{3}{1}\times \frac{12}{1} = 3 \times 12 = 36$ $\displaystyle 3 \div \frac{1}{12} = \frac{3}{1} \div \frac{1}{12} = \frac{3}{1}\times \frac{12}{1} = 3 \times 12 = 36$

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HSPT Math : Arithmetic

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #61 : Fractions

Example Question #56 : Identities And Properties

Example Question #1 : Fractions

Example Question #1 : How To Divide Fractions

Example Question #351 : Arithmetic

Example Question #352 : Arithmetic

Example Question #353 : Arithmetic

Example Question #354 : Arithmetic

Example Question #4 : Fractions

Example Question #355 : Arithmetic

All HSPT Math Resources

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