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PSAT Math : Arithmetic

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Example Questions

Example Question #1 : How To Find The Part From The Whole

A pitcher is currently $\dpi{80} \frac{4}{5}$ of the way full of water. If $\dpi{80} 3$ ounces are poured out, then the pitcher would be $\frac{3}{4}$ of the way full. How many ounces of water are in the pitcher before the $\dpi{80} 3$ ounces are poured out?

Possible Answers:

48oz

52oz

64oz

56oz

60oz

Correct answer:

48oz

Explanation:

Let p represent the total capacity of the pitcher. The current volume in the pitcher is $\dpi{80} \frac{4}{5}p$ . If this volume is depleted by $\dpi{80} 3$ ounces, it becomes $\dpi{80} \frac{3}{4}p$ . In other words:

$\dpi{80} \frac{4}{5}p - 3 = \frac{3}{4}p$

Solve this equation for $\dpi{80} p$ and get:

$\dpi{80} \frac{4}{5}p-\frac{3}{4}p=3$

$\dpi{80} \frac{16}{20}p-\frac{15}{20}p=3$

$\dpi{80} \frac{1}{20}p=3$

$\dpi{80} p=60$

Remember, $\dpi{80} p$ represents the total capacity of the pitcher, or 60 ounces. The question asks how many ounces are in the pitcher at the beginning of the problem, so evaluate $\dpi{80} \frac{4}{5}p=\frac{4}{5}\times 60=48$ .

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Example Question #1 : Operations With Fractions

Define an operation $\bigstar$ as follows:

For all real a, b ,

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

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

Possible Answers:

$-\frac{21}{50}$

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

$1\frac{11}{25}$

$\frac{7}{10}$

$-\frac{1}{10}$

Correct answer:

$\frac{7}{10}$

Explanation:

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

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

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

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

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

$= \frac{7}{10}$

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Example Question #2 : How To Subtract Fractions

Define an operation $\triangleright$ as follows:

For all real a, b ,

$a \triangleright b = 7 - ab$ .

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

Possible Answers:

$15 \frac{1}{3}$

$10\frac{2}{3}$

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

Correct answer:

$15 \frac{1}{3}$

Explanation:

$a \triangleright b = 7 - ab$

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

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

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

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

$= \frac{46}{3}$

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

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Example Question #3 : How To Subtract Fractions

Define an operation $\Cup$ as follows:

For all real a, b ,

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

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

Possible Answers:

$\frac{5}{7}$

$17\frac{13}{36}$

$\frac{1}{6}$

$3 \frac{13}{18}$

$4\frac{5}{18}$

Correct answer:

$3 \frac{13}{18}$

Explanation:

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

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

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

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

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

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

$= \frac{67}{18}$

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

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Example Question #4 : Operations With Fractions

Define an operation $\bigstar$ as follows:

For all real a, b ,

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

Evaluate $15 \bigstar 10$ .

Possible Answers:

$71\frac{2}{3}$

$4 \frac{1}{6}$

$46\frac{2}{3}$

Correct answer:

$4 \frac{1}{6}$

Explanation:

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

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

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

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

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

$= \frac{25}{6}$

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

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Example Question #2 : Operations With Fractions

Define an operation $\Cup$ as follows:

For all real a, b ,

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

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

Possible Answers:

$\frac{1}{5}$

$\frac{1}{10}$

$-\frac{41}{150}$

$\frac{7}{50}$

$-\frac{1}{50}$

Correct answer:

$\frac{1}{10}$

Explanation:

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

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

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

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

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

$= \frac{1}{10}$

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Example Question #1 : Operations With Fractions

Define an operation $\triangleright$ as follows:

For all real a, b ,

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

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

Possible Answers:

The correct answer is not among the other responses.

$-\frac{4}{5}$

$\frac{4}{5}$

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

$9\frac{3}{5}$

Correct answer:

The correct answer is not among the other responses.

Explanation:

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

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

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

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

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

$= \frac{16}{125}$

This answer is not among the choices given.

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Example Question #1661 : Psat Mathematics

Subtract and simplify the fractions:

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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

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Example Question #1 : How To Divide Fractions

Simplify:

Possible Answers:

a²/c²

a/b/c/d

ac/bd

ad/bc

It is already in simplest terms

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 #2 : 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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PSAT Math : Arithmetic

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Find The Part From The Whole

Example Question #1 : Operations With Fractions

Example Question #2 : How To Subtract Fractions

Example Question #3 : How To Subtract Fractions

Example Question #4 : Operations With Fractions

Example Question #2 : Operations With Fractions

Example Question #1 : Operations With Fractions

Example Question #1661 : Psat Mathematics

Example Question #1 : How To Divide Fractions

Example Question #2 : How To Divide Fractions

All PSAT Math Resources

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