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PSAT Math : PSAT Mathematics

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Example Questions

Example Question #1 : How To Divide Fractions

Define an operation $\odot$ as follows:

For all real numbers a ,b ,

$a \odot b = \frac{a}{b+2}$ .

Evaluate $\frac{3}{5} \odot \frac{1}{5}$ .

Possible Answers:

$3\frac{2}{3}$

$1\frac{8}{25}$

$\frac{25}{33}$

$\frac{3} {11}$

Correct answer:

$\frac{3} {11}$

Explanation:

$a \odot b = \frac{a}{b+2}$ ,

or, equivalently,

$a \odot b = a \div\left ( b+2 \right )$

$\frac{3}{5} \odot \frac{1}{5}= \frac{3}{5} \div \left (\frac{1}{5}+2 \right )$

$= \frac{3}{5} \div \frac{11}{5}$

$= \frac{3}{5} \cdot \frac{5}{11}$

$= \frac{3}{1} \cdot \frac{1}{11}$

$= \frac{3} {11}$

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Example Question #5 : How To Divide Fractions

Define an operation $\blacklozenge$ as follows:

For all real numbers a ,b ,

$a \; \blacklozenge \; b = a \div b^{2}$ .

Evaluate $25 \; \blacklozenge \; 6\frac{1}{4}$ .

Possible Answers:

100

$\frac{16} {25}$

$976\frac{9}{16}$

$\frac{4}{25}$

Correct answer:

$\frac{16} {25}$

Explanation:

$a \; \blacklozenge \; b = a \div b^{2}$

$25 \; \blacklozenge \; 6\frac{1}{4}= 25 \div \left ( 6\frac{1}{4} \right ) ^{2}$

$= 25 \div \left ( \frac{25}{4} \right ) ^{2}$

$= 25 \div \left ( \frac{625}{16} \right )$

$= \frac{25 }{1}\times \left ( \frac{16} {625}\right )$

$= \frac{1}{1}\times \left ( \frac{16} {25}\right )$

$= \frac{16} {25}$

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Example Question #12 : Operations With Fractions

Define an operation $\ominus$ as follows:

For all real numbers a ,b ,

$a \; \ominus \; b = a^{2} \div b$ .

Evaluate $16 \; \ominus \; 3\frac{1}{5}$ .

Possible Answers:

$1\frac{9}{16}$

$819\frac{1}{5}$

$\frac{5}{16}$

Correct answer:

Explanation:

$a \; \ominus \; b = a^{2} \div b$

$16 \; \ominus \; 3\frac{1}{5} = 16^{2} \div 3\frac{1}{5}$

$= 256 \div \frac{16}{5}$

$= \frac{256 }{1}\times \frac{5}{16}$

$= \frac{16}{1}\times \frac{5}{1}$

$= 16 \times 5 = 80$

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Example Question #7 : How To Divide Fractions

Define an operation $\amalg$ as follows:

For all real numbers a ,b ,

$a \amalg b = 5 - \frac{a}{b}$ .

Evaluate $1 \frac{1}{2}\; \amalg \; 2 \frac{1}{3}$ .

Possible Answers:

$3\frac{4}{9}$

$1\frac{1}{2}$

$4 \frac{5}{14}$

The correct answer is not among the other responses.

$4\frac{5}{7}$

Correct answer:

$4 \frac{5}{14}$

Explanation:

$a \amalg b = 5 - \frac{a}{b}$

or, equivalently,

$a \amalg b = 5 - a \div b$

$1 \frac{1}{2}\; \amalg \; 2 \frac{1}{3}= 5 - 1 \frac{1}{2} \div 2 \frac{1}{3}$

$= 5 - \frac{3}{2} \div \frac{7}{3}$

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

$= 5 - \frac{9}{14}$

$= 4 \frac{5}{14}$

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Example Question #111 : Fractions

Define an operation $\ddagger$ as follows:

For all real numbers a ,b ,

$a \ddagger b = \frac{a}{b} - \frac{b}{a}$ .

Evaluate $\frac{3}{4} \ddagger\frac{4}{3}$ .

Possible Answers:

$-3\frac{1}{12}$

$-1 \frac{31}{144}$

$1 \frac{31}{144}$

$3\frac{1}{12}$

Correct answer:

$-1 \frac{31}{144}$

Explanation:

$a \ddagger b = \frac{a}{b} - \frac{b}{a}$ ,

or, equivalently,

$a \ddagger b = (a \div b) - (b \div a)$

$\frac{3}{4} \ddagger\frac{4}{3} =\left (\frac{3}{4} \div \frac{4}{3} \right ) - \left (\frac{4}{3} \div \frac{3}{4} \right )$

$=\left (\frac{3}{4} \times \frac{3}{4} \right ) - \left (\frac{4}{3} \times \frac{4} {3}\right )$

$= \frac{9}{16} -\frac{16}{9}$

From here we need to find a common denominator.

$=\frac{9}{16}\bigg(\frac{9}{9}\bigg)- \frac{16}{9}\bigg(\frac{16}{16}\bigg)$

$= \frac{81}{144} -\frac{256}{144}$

$= -\frac{175}{144}$

$= -1 \frac{31}{144}$

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Example Question #111 : Fractions

Simplify:

$\frac{1+5}{\frac{6}{3+1}}$

Possible Answers:

$\frac{5}{8}$

$\frac{1}{4}$

$\frac{16}{3}$

Correct answer:

Explanation:

Begin by simplifying any additions that need to be done:

$\frac{1+5}{\frac{6}{3+1}}$

becomes

$\frac{6}{\frac{6}{4}}$

Now, remember that the numerator can be rewritten $\frac{6}{1}$ :

$\frac{\frac{6}{1}}{\frac{6}{4}}$

Now, when you divide fractions, you multiply the numerator by the reciprocal of the denominator:

$\frac{6}{1} * \frac{4}{6}$

Cancel the s and you get:

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Example Question #1671 : Psat Mathematics

If xy = 1 and 0 < x < 1, then which of the following must be true?

Possible Answers:

y = x

y = 1

y > 1

y < 1

y < x

Correct answer:

y > 1

Explanation:

If x is between 0 and 1, it must be a proper fraction (e.g., ½ or ¼). Solving the first equation for y, y = 1/x. When you divide 1 by a proper fraction between 0 and 1, the result is the reciprocal of that fraction, which will always be greater than 1.

To test this out, pick any fraction. Say x = ½. This makes y = 2.

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Example Question #602 : Arithmetic

Before going to school, Joey ran 1/3 of his daily total miles. In gym class, Joey did 2/3 of the remainder. What part of his daily total miles was left for after school?

Possible Answers:

4/9

7/9

2/9

2/3

1/3

Correct answer:

2/9

Explanation:

Before school, Joey did 1/3 of the total miles. In school, Joey did 2/3 of the remaining 2/3, or 4/9 of the running. When added to his in school run, his before school run of 3/9 brings his completed miles to 7/9 of his dialy total. Thus, only 2/9 of the total miles are left for after school.

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Example Question #1 : How To Multiply Fractions

Sally bought five computers for her office that cost $300, $405, $485, $520, and $555 respectively. She made a down payment of 2/5 the total cost and paid the rest in nine equal payments over the next nine months. Assuming no tax and no interest, what is the value of each of the nine payments?

Possible Answers:

351

1359

151

251

906

Correct answer:

151

Explanation:

The total cost of the 5 computers is 2265.

2/5 of 2265 = 906, which is what Sally pays up front.

2265 – 906 = 1359, which is what Sally still owes.

1359/9 = 151, which is the value of each of the 9 equal payments.

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Example Question #122 : Arithmetic

The price of a computer is reduced by 1/8. The new price is then reduced by 1/6. What fraction of the original price is the current price?

Possible Answers:

1/24

1/48

35/48

13/48

23/24

Correct answer:

35/48

Explanation:

Let the original price = p.

After the first reduction, the price is (7/8)p

After the second reduction, the price is (5/6)(7/8)p = (35/48)p

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PSAT Math : PSAT Mathematics

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Divide Fractions

Example Question #5 : How To Divide Fractions

Example Question #12 : Operations With Fractions

Example Question #7 : How To Divide Fractions

Example Question #111 : Fractions

Example Question #111 : Fractions

Example Question #1671 : Psat Mathematics

Example Question #602 : Arithmetic

Example Question #1 : How To Multiply Fractions

Example Question #122 : Arithmetic

All PSAT Math Resources

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