How to subtract fractions - ISEE Middle Level Quantitative Reasoning

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Question

$x = \frac{1}{2} + \frac{1}{3}$

$y = \frac{1}{2} + \frac{2}{3}$

Which is the greater quantity?

(a)

(b) $-\frac{1}{2}$

Answer

Subtract both sides of the two equations:

$x = \frac{1}{2} + \frac{1}{3}$

$y = \frac{1}{2} + \frac{2}{3}$

$x - y = \frac{1}{2} + \frac{1}{3}- \left ( \frac{1}{2} + \frac{2}{3} \right )$

$x - y = \frac{1}{2} + \frac{1}{3}-\frac{1}{2} - \frac{2}{3}$

$x - y = \frac{1}{2} -\frac{1}{2}+ \frac{1}{3} - \frac{2}{3}$

$x - y = - \frac{1}{3}$

Since $\frac{1}{3} < \frac{1}{2}$ , then $x-y = -\frac{1}{3} > - \frac{1}{2}$

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Question

Which is the greater quantity?

(a) $8 - 3 \frac{3}{7}$

(b) $5 \frac{3}{7}$

Answer

"Borrow" 1 from the 8 and subtract vertically:

$8 - 3 \frac{3}{7} = 7 \frac{7}{7} - 3 \frac{3}{7}$ :

$7 \frac{7}{7}$

$\underline{- 3 \frac{3}{7}}$

$4 \frac{4}{7}$

$4 \frac{4}{7} < 5 \frac{3}{7}$

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Question

Which is the greater quantity?

(a) $6 \frac{2}{7} - 2 \frac{6}{7}$

(b) $4 \frac{4}{7}$

Answer

"Borrow" 1 from the 6 and subtract vertically:

$6 \frac{2}{7} - 2 \frac{6}{7} = 5 \frac{9}{7} - 2 \frac{6}{7}$

$5 \frac{9}{7}$

$\underline{ - 2 \frac{6}{7}}$

$3 \frac{3}{7}$

$3 \frac{3}{7} < 4 \frac{4}{7}$

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Question

$b + 4 \frac{2}{3} = 9$

Which is the greater quantity?

(a)

(b)

Answer

(a)

(b) $b + 4 \frac{2}{3} = 9$

$b + 4 \frac{2}{3} - 4 \frac{2}{3} = 9- 4 \frac{2}{3}$

$b = 9- 4 \frac{2}{3} = 8\frac{3}{3}- 4\frac{2}{3} = 4\frac{1}{3}$

$4\frac{1}{3} = 4.333... < 4.5$ , so .

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Question

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

Answer

When subtracting fractions with different denominators, first change the fractions so that they have the same denominator. Do this by finding the least common multiple of both 4 and 8. Some multiples of 4 and 8 are:

4: 4, 8, 12, 16...

8: 8, 16, 24, 32...

Since 8 is the first common multiple between 4 and 8, change the fractions accordingly so that their denominators equal 8. Since $\tfrac{7}{8}$ already has a denominator of 8, it does not need to change. Change $\tfrac{3}{4}$ , however, accordingly.

$\frac{3}{4}\times\frac{2}{2}\rightarrow\frac{6}{8}$

The problem now looks like this:

$\frac{7}{8}-\frac{6}{8}=$

Subtract the numerators. The result is your answer.

$\frac{7}{8}-\frac{6}{8}=\frac{7-6}{8}=\frac{1}{8}$

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Question

$2\tfrac{1}{2} - 1\tfrac{3}{5} =$

Answer

When dealing with fractions and mixed numbers, first convert the mixed numbers to improper fractions.

$2\tfrac{1}{2} \rightarrow\tfrac{5}{2}$

$1\tfrac{3}{5}\rightarrow\tfrac{8}{5}$

The next thing you must do is change the denominators so that they are equal. Do this by finding the least common multiple of 2 and 5. Some multiples of 2 and 5 are:

2: 2, 4, 6, 8, 10...

5: 5, 10, 15, 20...

10 is the first common multiple of both 2 and 5, so change the fractions accordingly so that their denominators both equal 10.

$\tfrac{5}{2}\times\tfrac{5}{5}\rightarrow\tfrac{25}{10}$

$\tfrac{8}{5}\times\tfrac{2}{2}\rightarrow\tfrac{16}{10}$

The problem now looks like this:

$\tfrac{25}{10}-\tfrac{16}{10}=$

Subtract the numerators of the fraction. The result is your answer.

$\tfrac{25}{10}-\tfrac{16}{10}=\tfrac{25-16}{10}=\tfrac{9}{10}$

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Question

$\frac{3}{4}-\frac{2}{3}$

Answer

$\frac{3}{4}-\frac{2}{3}$

In order to solve this problem, we first have to find common denominators.

$\frac{3}{4}\times\frac{3}{3}=\frac{9}{12}$

$\frac{2}{3\time}\times\frac{4}{4}=\frac{8}{12}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{9}{12}-\frac{8}{12}=\frac{1}{12}$

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Question

$\frac{8}{12}-\frac{1}4{}$

Answer

$\frac{8}{12}-\frac{1}4{}$

In order to solve this problem, we first have to find common denominators.

$\frac{1}{4}\times\frac{3}{3}=\frac{3}{12}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{8}{12}-\frac{3}{12}=\frac{5}{12}$

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Question

Solve:

$\frac{2}{3}-\frac{3}{5}$

Answer

$\frac{2}{3}-\frac{3}{5}$

In order to solve this problem, we first have to find common denominators.

$\frac{2}{3}\times\frac{5}{5}=\frac{10}{15}$

$\frac{3}{5}\times \frac{3}{3}=\frac{9}{15}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{10}{15}-\frac{9}{15}=\frac{1}{15}$

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Question

Solve:

$\frac{7}{8}-\frac{3}{16}$

Answer

$\frac{7}{8}-\frac{3}{16}$

In order to solve this problem, we first have to find common denominators.

$\frac{7}{8}\times\frac{2}{2}=\frac{14}{16}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{14}{16}-\frac{3}{16}=\frac{11}{16}$

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Question

$\frac{5}{6}-\frac{1}{4}$

Answer

$\frac{5}{6}-\frac{1}{4}$

In order to solve this problem, we first have to find common denominators.

$\frac{5}{6}\times\frac{2}{2}=\frac{10}{12}$

$\frac{1}{4}\times\frac{3}{3}=\frac{3}{12}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{10}{12}-\frac{3}{12}=\frac{7}{12}$

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Question

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

Answer

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

In order to solve this problem, we first have to find common denominators.

$\frac{3}{5}\times\frac{2}{2}=\frac{6}{10}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{9}{10}-\frac{6}{10}=\frac{3}{10}$

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Question

$\frac{1}{2}-\frac{1}{3}$

Answer

$\frac{1}{2}-\frac{1}{3}$

In order to solve this problem, we first have to find common denominators.

$\frac{1}{2}\times\frac{3}{3}=\frac{3}{6}$

$\frac{1}{3}\times\frac{2}{2}=\frac{2}{6}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{3}{6}-\frac{2}{6}=\frac{1}{6}$

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Question

$\frac{3}{4}-\frac{5}{8}$

Answer

$\frac{3}{4}-\frac{5}{8}$

In order to solve this problem, we first have to find common denominators.

$\frac{3}{4}\times\frac{2}{2}=\frac{6}{8}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{6}{8}-\frac{5}{8}=\frac{1}{8}$

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Question

Solve the following:

$\frac{2}{3}-\frac{1}{8}$

Answer

$\frac{2}{3}-\frac{1}{8}$

In order to solve this problem, we first have to find common denominators.

$\frac{2}{3}\times\frac{8}{8}=\frac{16}{24}$

$\frac{1}{8}\times\frac{3}{3}=\frac{3}{24}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{16}{24}-\frac{3}{24}=\frac{13}{24}$

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Question

$\frac{9}{11}-\frac{3}{4}$

Answer

$\frac{9}{11}-\frac{3}{4}$

In order to solve this problem, we first have to find common denominators.

$\frac{9}{11}\times\frac{4}{4}=\frac{36}{44}$

$\frac{3}{4}\times\frac{11}{11}=\frac{33}{44}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{36}{44}-\frac{33}{44}=\frac{3}{44}$

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Question

$\frac{8}{9}-\frac{1}{8}$

Answer

$\frac{8}{9}-\frac{1}{8}$

In order to solve this problem, we first have to find common denominators.

$\frac{8}{9}\times\frac{8}{8}=\frac{64}{72}$

$\frac{1}{8}\times\frac{9}{9}=\frac{9}{72}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{64}{72}-\frac{9}{72}=\frac{55}{72}$

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Question

$\frac{4}{5}-\frac{2}{7}$

Answer

$\frac{4}{5}-\frac{2}{7}$

In order to solve this problem, we first have to find common denominators.

$\frac{4}{5}\times\frac{7}{7}=\frac{28}{35}$

$\frac{2}{7}\times\frac{5}{5}=\frac{10}{35}$

Now that we have common denominators, we can subtract the fractions. Remember, when we add and subtract fractions, we only add or subtract the numerator.

$\frac{28}{35}-\frac{10}{35}=\frac{18}{35}$

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Question

Jessica ate $\frac{1}{3}$ of the cake and Megan ate $\frac{1}{2}$ . How much more of the cake did Megan eat?

Answer

$\frac{1}2{}-\frac{1}{3}$

In order to solve this problem, we first need to make common denominators.

$\frac{1}3{\times\frac{2}{2}=\frac{2}{6}}$

$\frac{1}{2}\times\frac{3}{3}=\frac{3}{6}$

Now that we have common denominators, we can subtract the fractions. Remember, when we subtract fractions, the denominator stays the same, we only subtract the numerator.

$\frac{3}{6}-\frac{2}{6}=\frac{1}{6}$

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Question

Tim mowed $\frac{1}{7}$ of the yard and Tom mowed $\frac{1}{3}$ . How much more of the yard did Tom mow?

Answer

$\frac{1}{3}-\frac{1}{7}$

In order to solve this problem, we first need to make common denominators.

$\frac{1}{7}\times\frac{3}{3}=\frac{3}{21}$

$\frac{1}{3}\times \frac{7}{7}=\frac{7}{21}$

Now that we have common denominators, we can subtract the fractions. Remember, when we subtract fractions, the denominator stays the same, we only subtract the numerator.

$\frac{7}{21}-\frac{3}{21}=\frac{4}{21}$

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