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PSAT Math : Operations with Fractions

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Example Questions

Example Question #102 : Arithmetic

Define an operation $\bigstar$ $\displaystyle \bigstar$ as follows:

For all real a, b $\displaystyle a, b$,

$a \bigstar b = 2a - 3b$ $\displaystyle a \bigstar b = 2a - 3b$.

Evaluate $\frac{4}{5} \bigstar \frac{3}{10}$ $\displaystyle \frac{4}{5} \bigstar \frac{3}{10}$.

Possible Answers:

$-1\frac{4}{5}$ $\displaystyle -1\frac{4}{5}$

$-\frac{21}{50}$ $\displaystyle -\frac{21}{50}$

$1\frac{11}{25}$ $\displaystyle 1\frac{11}{25}$

$\frac{7}{10}$ $\displaystyle \frac{7}{10}$

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

Correct answer:

$\frac{7}{10}$ $\displaystyle \frac{7}{10}$

Explanation:

$a \bigstar b = 2a - 3b$ $\displaystyle a \bigstar b = 2a - 3b$

$\frac{4}{5} \bigstar \frac{3}{10} = 2 \cdot \frac{4}{5} - 3 \cdot \frac{3}{10}$ $\displaystyle \frac{4}{5} \bigstar \frac{3}{10} = 2 \cdot \frac{4}{5} - 3 \cdot \frac{3}{10}$

$= \frac{2}{1} \cdot \frac{4}{5} - \frac{3}{1} \cdot \frac{3}{10}$ $\displaystyle = \frac{2}{1} \cdot \frac{4}{5} - \frac{3}{1} \cdot \frac{3}{10}$

$= \frac{8}{5} - \frac{9}{10}$ $\displaystyle = \frac{8}{5} - \frac{9}{10}$

$= \frac{16}{10} - \frac{9}{10}$ $\displaystyle = \frac{16}{10} - \frac{9}{10}$

$= \frac{7}{10}$ $\displaystyle = \frac{7}{10}$

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Example Question #102 : Fractions

Define an operation $\triangleright$ $\displaystyle \triangleright$ as follows:

For all real a, b $\displaystyle a, b$,

$a \triangleright b = 7 - ab$ $\displaystyle a \triangleright b = 7 - ab$.

Evaluate $5 \triangleright \left ( - \frac{5}{3}\right )$ $\displaystyle 5 \triangleright \left ( - \frac{5}{3}\right )$.

Possible Answers:

$\displaystyle 11$

$15 \frac{1}{3}$ $\displaystyle 15 \frac{1}{3}$

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

$10\frac{2}{3}$ $\displaystyle 10\frac{2}{3}$

$\displaystyle 4$

Correct answer:

$15 \frac{1}{3}$ $\displaystyle 15 \frac{1}{3}$

Explanation:

$a \triangleright b = 7 - ab$ $\displaystyle a \triangleright b = 7 - ab$

$5 \triangleright \left ( - \frac{5}{3}\right ) = 7 - 5 \left ( - \frac{5}{3}\right )$ $\displaystyle 5 \triangleright \left ( - \frac{5}{3}\right ) = 7 - 5 \left ( - \frac{5}{3}\right )$

$= 7 - \frac{5}{1} \left ( - \frac{5}{3}\right )$ $\displaystyle = 7 - \frac{5}{1} \left ( - \frac{5}{3}\right )$

$= 7 - \left ( - \frac{25}{3}\right )$ $\displaystyle = 7 - \left ( - \frac{25}{3}\right )$

$= \frac{21}{3} + \frac{25}{3}$ $\displaystyle = \frac{21}{3} + \frac{25}{3}$

$= \frac{46}{3}$ $\displaystyle = \frac{46}{3}$

$=15 \frac{1}{3}$ $\displaystyle =15 \frac{1}{3}$

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Example Question #2 : How To Subtract Fractions

Define an operation $\Cup$ $\displaystyle \Cup$ as follows:

For all real a, b $\displaystyle a, b$,

$a \Cup b = a - b^{2}$ $\displaystyle a \Cup b = a - b^{2}$

Evaluate $5\frac{1}{2} \Cup 1 \frac{1}{3}$ $\displaystyle 5\frac{1}{2} \Cup 1 \frac{1}{3}$.

Possible Answers:

$\frac{1}{6}$ $\displaystyle \frac{1}{6}$

$3 \frac{13}{18}$ $\displaystyle 3 \frac{13}{18}$

$17\frac{13}{36}$ $\displaystyle 17\frac{13}{36}$

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

$4\frac{5}{18}$ $\displaystyle 4\frac{5}{18}$

Correct answer:

$3 \frac{13}{18}$ $\displaystyle 3 \frac{13}{18}$

Explanation:

$a \Cup b = a - b^{2}$ $\displaystyle a \Cup b = a - b^{2}$

$5\frac{1}{2} \Cup 1 \frac{1}{3} = 5\frac{1}{2} -\left ( 1 \frac{1}{3} \right )^{2}$ $\displaystyle 5\frac{1}{2} \Cup 1 \frac{1}{3} = 5\frac{1}{2} -\left ( 1 \frac{1}{3} \right )^{2}$

$= \frac{11}{2} -\left ( \frac{4}{3} \right )^{2}$ $\displaystyle = \frac{11}{2} -\left ( \frac{4}{3} \right )^{2}$

$= \frac{11}{2} -\left ( \frac{4^{2}}{3^{2}} \right )$ $\displaystyle = \frac{11}{2} -\left ( \frac{4^{2}}{3^{2}} \right )$

$= \frac{11}{2} - \frac{16}{9}$ $\displaystyle = \frac{11}{2} - \frac{16}{9}$

$= \frac{99}{18} - \frac{32}{18}$ $\displaystyle = \frac{99}{18} - \frac{32}{18}$

$= \frac{67}{18}$ $\displaystyle = \frac{67}{18}$

$=3 \frac{13}{18}$ $\displaystyle =3 \frac{13}{18}$

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Example Question #2 : How To Subtract Fractions

Define an operation $\bigstar$ $\displaystyle \bigstar$ as follows:

For all real a, b $\displaystyle a, b$,

$a \bigstar b = \frac{1}{2} a - \frac{1}{3} b$ $\displaystyle a \bigstar b = \frac{1}{2} a - \frac{1}{3} b$.

Evaluate $15 \bigstar 10$ $\displaystyle 15 \bigstar 10$.

Possible Answers:

$46\frac{2}{3}$ $\displaystyle 46\frac{2}{3}$

$71\frac{2}{3}$ $\displaystyle 71\frac{2}{3}$

$\displaystyle 10$

$\displaystyle 0$

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

Correct answer:

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

Explanation:

$a \bigstar b = \frac{1}{2} a - \frac{1}{3} b$ $\displaystyle a \bigstar b = \frac{1}{2} a - \frac{1}{3} b$

$15 \bigstar 10 = \frac{1}{2} \cdot 15 - \frac{1}{3} \cdot 10$ $\displaystyle 15 \bigstar 10 = \frac{1}{2} \cdot 15 - \frac{1}{3} \cdot 10$

$= \frac{1}{2} \cdot \frac{15 }{1}- \frac{1}{3} \cdot \frac{10}{1}$ $\displaystyle = \frac{1}{2} \cdot \frac{15 }{1}- \frac{1}{3} \cdot \frac{10}{1}$

$= \frac{15}{2} - \frac{10}{3}$ $\displaystyle = \frac{15}{2} - \frac{10}{3}$

$= \frac{45}{6} - \frac{20}{6}$ $\displaystyle = \frac{45}{6} - \frac{20}{6}$

$= \frac{25}{6}$ $\displaystyle = \frac{25}{6}$

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

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Example Question #2 : Operations With Fractions

Define an operation $\Cup$ $\displaystyle \Cup$ as follows:

For all real a, b $\displaystyle a, b$,

$a \Cup b = \frac{3}{5} a -b$ $\displaystyle a \Cup b = \frac{3}{5} a -b$

Evaluate $\frac{1}{3 }\Cup \frac{1}{10}$ $\displaystyle \frac{1}{3 }\Cup \frac{1}{10}$

Possible Answers:

$\frac{1}{5}$ $\displaystyle \frac{1}{5}$

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

$-\frac{41}{150}$ $\displaystyle -\frac{41}{150}$

$\frac{7}{50}$ $\displaystyle \frac{7}{50}$

$-\frac{1}{50}$ $\displaystyle -\frac{1}{50}$

Correct answer:

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

Explanation:

$a \Cup b = \frac{3}{5} a -b$ $\displaystyle a \Cup b = \frac{3}{5} a -b$

$\frac{1}{3 }\Cup \frac{1}{10} = \frac{3}{5}\left (\frac{1}{3} \right ) - \frac{1}{10}$ $\displaystyle \frac{1}{3 }\Cup \frac{1}{10} = \frac{3}{5}\left (\frac{1}{3} \right ) - \frac{1}{10}$

$= \frac{1}{5}\left (\frac{1}{1} \right ) - \frac{1}{10}$ $\displaystyle = \frac{1}{5}\left (\frac{1}{1} \right ) - \frac{1}{10}$

$= \frac{1}{5} - \frac{1}{10}$ $\displaystyle = \frac{1}{5} - \frac{1}{10}$

$= \frac{2}{10} - \frac{1}{10}$ $\displaystyle = \frac{2}{10} - \frac{1}{10}$

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

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Example Question #1 : Operations With Fractions

Define an operation $\triangleright$ $\displaystyle \triangleright$ as follows:

For all real a, b $\displaystyle a, b$,

$a \triangleright b = a^{2}- b^{3}$ $\displaystyle a \triangleright b = a^{2}- b^{3}$.

Evaluate $\frac{4}{5} \triangleright \frac{4}{5}$ $\displaystyle \frac{4}{5} \triangleright \frac{4}{5}$.

Possible Answers:

The correct answer is not among the other responses.

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

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

$-9\frac{3}{5}$ $\displaystyle -9\frac{3}{5}$

$9\frac{3}{5}$ $\displaystyle 9\frac{3}{5}$

Correct answer:

The correct answer is not among the other responses.

Explanation:

$a \triangleright b = a^{2}- b^{3}$ $\displaystyle a \triangleright b = a^{2}- b^{3}$

$\frac{4}{5} \triangleright \frac{4}{5} =\left ( \frac{4}{5} \right )^{2} - \left ( \frac{4}{5} \right )^{3}$ $\displaystyle \frac{4}{5} \triangleright \frac{4}{5} =\left ( \frac{4}{5} \right )^{2} - \left ( \frac{4}{5} \right )^{3}$

$=\left ( \frac{4^{2}}{5^{2}} \right ) - \left ( \frac{4^{3}}{5^{3}} \right )$ $\displaystyle =\left ( \frac{4^{2}}{5^{2}} \right ) - \left ( \frac{4^{3}}{5^{3}} \right )$

$= \frac{16}{25} - \frac{64}{125}$ $\displaystyle = \frac{16}{25} - \frac{64}{125}$

$= \frac{80}{125} - \frac{64}{125}$ $\displaystyle = \frac{80}{125} - \frac{64}{125}$

$= \frac{16}{125}$ $\displaystyle = \frac{16}{125}$

This answer is not among the choices given.

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Example Question #1 : How To Subtract Fractions

Subtract and simplify the fractions:

$\small \frac{7}{5} - \frac{3}{6}$ $\displaystyle \small \frac{7}{5} - \frac{3}{6}$

Possible Answers:

$\small \frac{9}{10}$ $\displaystyle \small \frac{9}{10}$

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

$\small \small \frac{7}{10}$ $\displaystyle \small \small \frac{7}{10}$

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

Correct answer:

$\small \frac{9}{10}$ $\displaystyle \small \frac{9}{10}$

Explanation:

First we need to give them common denominators. Remember, whatever you do to the denominator you must do to the numerator:

$\small \small \frac{6}{6} \times \frac{7}{5} -\frac{5}{5} \times \frac{3}{6}$ $\displaystyle \small \small \frac{6}{6} \times \frac{7}{5} -\frac{5}{5} \times \frac{3}{6}$

$\small = \frac{42}{30}- \frac{15}{30}$ $\displaystyle \small = \frac{42}{30}- \frac{15}{30}$

From here, we do algebraic operations to subtract and simplify.

$\small = \frac{42 - 15}{30}$ $\displaystyle \small = \frac{42 - 15}{30}$

$\small = \frac{27}{30}$ $\displaystyle \small = \frac{27}{30}$

$\small = \frac{9}{10}$ $\displaystyle \small = \frac{9}{10}$

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Example Question #102 : Fractions

Simplify:

Possible Answers:

It is already in simplest terms

ac/bd

ad/bc

a²/c²

a/b/c/d

Correct answer:

ad/bc

Explanation:

Division is the same as multiplying by the reciprocal. Thus, a/b ÷ c/d = a/b x d/c = ad/bc

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Example Question #1 : How To Divide Fractions

If p is a positive integer, and 4 is the remainder when p-8 is divided by 5, which of the following could be the value of p?

Possible Answers:

Correct answer:

Explanation:

Remember that if x has a remainder of 4 when divided by 5, x minus 4 must be divisible by 5. We are therefore looking for a number p such that p - 8 - 4 is divisible by 5. The only answer choice that fits this description is 17.

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Example Question #4 : Operations With Fractions

If $\dpi{100} \small x=\frac{2}{3}$ $\displaystyle \dpi{100} \small x=\frac{2}{3}$ and $\dpi{100} \small y= \frac {3}{4}$ $\displaystyle \dpi{100} \small y= \frac {3}{4}$, then what is the value of $\dpi{100} \small \frac {x}{y}$ $\displaystyle \dpi{100} \small \frac {x}{y}$?

Possible Answers:

$\dpi{100} \small 2$ $\displaystyle \dpi{100} \small 2$

$\dpi{100} \small \frac{9}{8}$ $\displaystyle \dpi{100} \small \frac{9}{8}$

$\dpi{100} \small \frac{8}{9}$ $\displaystyle \dpi{100} \small \frac{8}{9}$

$\dpi{100} \small \frac{1}{2}$ $\displaystyle \dpi{100} \small \frac{1}{2}$

$\dpi{100} \small \frac{5}{7}$ $\displaystyle \dpi{100} \small \frac{5}{7}$

Correct answer:

$\dpi{100} \small \frac{8}{9}$ $\displaystyle \dpi{100} \small \frac{8}{9}$

Explanation:

Dividing by a number (in this case $\dpi{100} \small \frac {3}{4}$ $\displaystyle \dpi{100} \small \frac {3}{4}$) is equivalent to multiplying by its reciprocal (in this case $\dpi{100} \small \frac {4}{3}$ $\displaystyle \dpi{100} \small \frac {4}{3}$). Therefore:

$\dpi{100} \small \frac {2}{3}\div \frac{3}{4} = \frac{2}{3}\times \frac{4}{3} = \frac{8}{9}$ $\displaystyle \dpi{100} \small \frac {2}{3}\div \frac{3}{4} = \frac{2}{3}\times \frac{4}{3} = \frac{8}{9}$

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PSAT Math : Operations with Fractions

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #102 : Arithmetic

Example Question #102 : Fractions

Example Question #2 : How To Subtract Fractions

Example Question #2 : How To Subtract Fractions

Example Question #2 : Operations With Fractions

Example Question #1 : Operations With Fractions

Example Question #1 : How To Subtract Fractions

Example Question #102 : Fractions

Example Question #1 : How To Divide Fractions

Example Question #4 : Operations With Fractions

All PSAT Math Resources

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