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PSAT Math : Arithmetic

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Example Questions

Example Question #1 : How To Find A Solution To A Compound Fraction

Simplify:

$\frac{\frac{3}{4}-\frac{2}{5}}{2- \frac{9}{10}}$

Possible Answers:

$\frac{7}{22}$

$\frac{77}{200}$

$\frac{147}{200}$

$\frac{5}{72}$

$\frac{7}{38}$

Correct answer:

$\frac{7}{22}$

Explanation:

$\frac{\frac{3}{4}-\frac{2}{5}}{2- \frac{9}{10}}$

$= \left ( \frac{3}{4}-\frac{2}{5} \right ) \div \left ( 2- \frac{9}{10} \right )$

$= \left ( \frac{15}{20}-\frac{8}{20} \right ) \div \left ( \frac{20}{10}- \frac{9}{10} \right )$

$= \frac{7}{20} \div \frac{11}{10}$

$= \frac{7}{20} \cdot \frac{10}{11}$

$= \frac{7}{2} \cdot \frac{1}{11}$

$= \frac{7}{22}$

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Example Question #3 : How To Find A Solution To A Compound Fraction

Simplify:

$\frac{1-\frac{1}{4}}{1-\frac{1}{8}}$

Possible Answers:

$\frac{21}{32}$

$\frac{6}{7}$

$\frac{8}{9}$

$\frac{1}{2}$

$\frac{7}{8}$

Correct answer:

$\frac{6}{7}$

Explanation:

$\frac{1-\frac{1}{4}}{1-\frac{1}{8}}$

$= \left (1 -\frac{1}{4} \right ) \div \left (1-\frac{1}{8} \right )$

$= \left (\frac{4}{4}-\frac{1}{4} \right ) \div \left (\frac{8}{8}-\frac{1}{8} \right )$

$= \frac{3}{4} \div \frac{7}{8}$

$= \frac{3}{4} \cdot \frac{8}{7}$

$= \frac{3}{1} \cdot \frac{2}{7}$

$= \frac{6}{7}$

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Example Question #2 : How To Find A Solution To A Compound Fraction

Simplify:

$\frac{2-\frac{1}{4}}{1+\frac{1}{7}}$

Possible Answers:

$1 \frac{17}{32}$

None of the other responses gives the correct answer.

$1\frac{1}{28}$

$1\frac{27}{28}$

Correct answer:

$1 \frac{17}{32}$

Explanation:

$\frac{2-\frac{1}{4}}{1+\frac{1}{7}}$

$= \left ( 2-\frac{1}{4} \right ) \div \left ( 1+\frac{1}{7} \right )$

$= \left (\frac{8}{4}-\frac{1}{4} \right ) \div \left ( \frac{7}{7}+\frac{1}{7} \right )$

$= \frac{7}{4} \div \frac{8}{7}$

$= \frac{7}{4} \cdot \frac{7}{8}$

$= \frac{49}{32}$

$= 1 \frac{17}{32}$

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Example Question #6 : General Fractions

Simplify:

$\frac{\frac{1}{3}+\frac{4}{5} }{\frac{2}{3}+\frac{4}{5} }$

Possible Answers:

$\frac{1}{2}$

$\frac{17} {22}$

$\frac{2}{9}$

$\frac{374}{225}$

$\frac{5}{8}$

Correct answer:

$\frac{17} {22}$

Explanation:

$\frac{\frac{1}{3}+\frac{4}{5} }{\frac{2}{3}+\frac{4}{5} }$

$=\left ( \frac{1}{3}+\frac{4}{5} \right ) \div\left ( \frac{2}{3}+\frac{4}{5} \right )$

$=\left ( \frac{5}{15}+\frac{12}{15} \right ) \div\left ( \frac{10}{15}+\frac{12}{15} \right )$

$= \frac{17}{15} \div \frac{22}{15}$

$= \frac{17}{15} \cdot \frac{15}{22}$

$= \frac{17}{1} \cdot \frac{1}{22}$

$= \frac{17} {22}$

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Example Question #3 : How To Find A Solution To A Compound Fraction

Simplify:

$\frac{\frac{1}{2}-\frac{1}{10} }{\frac{1}{4}-\frac{1}{5} }$

Possible Answers:

$\frac{1}{50}$

$\frac{1}{2}$

The correct answer is not given among the other responses.

Correct answer:

The correct answer is not given among the other responses.

Explanation:

$\frac{\frac{1}{2}-\frac{1}{10} }{\frac{1}{4}-\frac{1}{5} }$

$= \left ( \frac{1}{2}-\frac{1}{10} \right )\div \left ( \frac{1}{4}-\frac{1}{5} \right )$

$= \left ( \frac{5}{10}-\frac{1}{10} \right )\div \left ( \frac{5}{20}-\frac{4}{20} \right )$

$= \frac{4}{10} \div \frac{1}{20}$

$= \frac{4}{10} \cdot \frac{20}{1}$

$= \frac{4}{1} \cdot \frac{2}{1}$

$= \frac{8}{1} = 8$

This answer is not among the given choices.

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Example Question #1 : How To Find Out A Mixed Fraction From An Improper Fraction

To make 48 cookies you need 1 cup white sugar, 1 cup packed brown sugar and 3 cups of flour. You want to make 12 cookies, so you adjust the ingredients accordingly. After the adjustment, how many total cups of dry ingredients do you have?

Possible Answers:

¹/₄ c

1 ¹/₂ c

1 ¹/₄ c

³/₄ c

1 ³/₄ c

Correct answer:

1 ¹/₄ c

Explanation:

Going from 48 cookies to 12 cookies is a scaling factor of ¹/₄, so all the ingredients get multiplied by ¹/₄. In order to make 12 cookies you will need ¹/₄ c white sugar, 1/4 c packed brown sugar, and ³/₄ c flour. Added all together you get ⁵/₄ or 1 ¹/₄ c.

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Example Question #2 : How To Find Out A Mixed Fraction From An Improper Fraction

What is $\frac{17}{12}$ written as a mixed number?

Possible Answers:

$2\frac{1}{6}$

$\frac{7}{12}$

$1\frac{1}{3}$

$1\frac{5}{12}$

$3\frac{1}{2}$

Correct answer:

$1\frac{5}{12}$

Explanation:

How many times does 12 go into 17? Once with a remainder of 5.

So $\frac{17}{12}$ becomes $1\frac{5}{12}$

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Example Question #1 : How To Find Out A Mixed Fraction From An Improper Fraction

If 3x=20 , what does equal?

Possible Answers:

$8\tfrac{1}{3}$

$6\tfrac{1}{3}$

$7\tfrac{2}{3}$

$7\tfrac{1}{3}$

$6\tfrac{2}{3}$

Correct answer:

$6\tfrac{2}{3}$

Explanation:

First, you must solve the equation for . Then, you must convert the answer's improper fraction to a mixed fraction.

3x=2

Solve for .

$x = \frac{20}{3}$

Now, to convert an improper fraction, you must see how many times the denominator goes into the numerator, and then leave the remainder as a fraction. Example below:

Find out how many times the denominator goes into the numerator:

$20\div3 = 6, remainder 2$

Therefore,

$x = 6\tfrac{2}{3}$

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Example Question #1 : How To Find Out An Improper Fraction From A Mixed Fraction

What is $3\frac{2}{5}$ as an improper fraction?

Possible Answers:

$\frac{21}{5}$

$\frac{11}{5}$

$\frac{10}{2}$

$\frac{17}{5}$

$\frac{6}{2}$

Correct answer:

$\frac{17}{5}$

Explanation:

$3\frac{2}{5}$ becomes $3 + \frac{2}{5} = \frac{15}{5} + \frac{2}{5} = \frac{17}{5}$

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Example Question #1 : How To Find The Amount Of Rational Numbers Between Two Numbers

Two numbers have a greatest common divisor of 4 and a least common multiple of 40. How many different pairs of numbers are there that satisfy these properties?

Possible Answers:

$Infinitely\ many$

Correct answer:

Explanation:

The greatest common divisor is 4. This means that both numbers must be divisible by 4. Furthermore, the least common multiple is 40, so both must divide 40.

The prime factorization of 40 is $2\cdot 2\cdot 2\cdot 5$ . For a number to divide 40, it must therefore be composed of (at most) three 2's and one 5. Because $4=2\cdot 2$ divides both numbers, we also know that they must both have at least two 2's.

Now each number will have either two or three 2's and zero or one 5's. However, we also know that they can't both have three 2's, (since then the greatest common divisor would have three 2's as well). Similarly, only one can have a 5.

In essence, our problem becomes one of choice. We have 2 places with value 4. We choose to give a 5 to one of the two. We then give a 2 to one of the two. If we give a 5 and a 2 to the same side, we end up with $5\cdot 2\cdot 4=40$ and 4. If we give a 5 to one and a 2 to the other, we end up with $5\cdot 4=20$ and $2\cdot 4=8$ .

Thus our two pairs are:

4,40 and 8,20

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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 A Solution To A Compound Fraction

Example Question #3 : How To Find A Solution To A Compound Fraction

Example Question #2 : How To Find A Solution To A Compound Fraction

Example Question #6 : General Fractions

Example Question #3 : How To Find A Solution To A Compound Fraction

Example Question #1 : How To Find Out A Mixed Fraction From An Improper Fraction

Example Question #2 : How To Find Out A Mixed Fraction From An Improper Fraction

Example Question #1 : How To Find Out A Mixed Fraction From An Improper Fraction

Example Question #1 : How To Find Out An Improper Fraction From A Mixed Fraction

Example Question #1 : How To Find The Amount Of Rational Numbers Between Two Numbers

All PSAT Math Resources

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