Manipulation of Fractions - Basic Arithmetic

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Question

What is $1\tfrac{2}{3}+3\tfrac{2}{3}$ in fraction form?

Answer

To rewrite the mixed fractions as improper fractions we multiply the whole number by the denominator and add the numerator to get our new numerator. Our denominator stays the same. With this in mind we get the following:

$1\tfrac{2}{3}= \frac{(1 \cdot 3) +2}{3}=\frac{3+2}{3}=\frac{5}{3}$ and

$3\tfrac{2}{3}=\frac{(3 \cdot 3)+2}{3}=\frac{9+2}{3}=\frac{11}{3}$ .

Adding them gives us

$\frac{5}{3}+\frac{11}{3}=\frac{16}{3}$ .

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Question

Convert this mixed number $3\frac{1}{4}$ to an improper fraction.

Answer

The correct answer is $\frac{13}{4}$ .

The mixed number can be broken down as follows: 3 is the whole number, 1 is the numerator, and 4 is the denominator.

In order to convert the mixed number to an improper fraction, you must first multiply the denominator with the whole number and then add the numerator to it.

This answer is the numerator of the improper fraction:

$4\cdot 3+1 = 12+1 = 13$ .

The denominator of the fraction remains the same and the final answer is $\frac{13}{4}$ .

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Question

Find the sum.

$9\frac{2}{3}+4\frac{1}{27}= ?$

Answer

To add mixed numbers, you need to first change both of them into improper fractions.

$9\frac{2}{3}=\frac{29}{3}$

$4\frac{1}{27}=\frac{109}{27}$

Then, add the two improper fractions like you would any other fraction.

We need both fractions to have the same denominator before we can add them. Since 27 is a multiple of 3, we only need to change 1 fraction.

$\frac{29}{3}\times\frac{9}{9}=\frac{261}{27}$

Now, we can add these fractions.

$\frac{261}{27}+\frac{109}{27}=\frac{370}{27}$

Finally, change the improper fraction back into a mixed number.

$\frac{370}{27}=13\frac{19}{27}$

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Question

What is $\frac{75}{6}$ equivalent to?

Answer

This question is asking you to change an improper fraction into a mixed number. To do that we will need to divide 75 by 6.

$75\div6=12 \text{ remainder }3$

Now, we place the remainder as the numerator of a fraction with a denominator that is 6 because we just divided by six.

So then, 12 remainder 3 becomes $12\frac{3}{6}$ .

Simplify this mixed fraction.

$12\frac{3}{6}=12\frac{1}{2}$

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Question

Express the following fraction as a decimal:

$\frac{18}{30}$

Answer

To easily express a fraction as a decimal, we need to convert the denominator into a power of . We can do this by dividing both the top and the bottom of the given fraction by to express it in tenths.

$\frac{18}{30}\div \frac{3}{3}= \frac{6}{10}$

$\frac{6}{10}$ , or six tenths, can be expressed as .

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Question

What is the decimal equivalent to $\frac{3}{4}$ ?

Answer

The correct answer is

$\frac{3}{4}$ is greater than half so you can eliminate and right away.

You should memorize common fractions:

$\frac{3}{4}$

$\frac{2}{3}$

$\frac{1}{6}$

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Question

Which fraction is less than $\frac{2}{3}$ ?

Answer

The best way to answer this question is to find the smallest fraction among the answer choices.

Comparing fractions can be done easily when they are converted to decimals.

$\frac{2}{3}=2\div3=0.667$

$\frac{6}{7}=6\div7=0.857$

$\frac{8}{11}=8\div11=0.727$

$\frac{17}{23}=17\div23=0.739$

$\frac{18}{23}=18\div23=0.783$

$\frac{3}{5}=3\div5=0.60$

Of these options, only ${\frac{3}{5}}$ , or 0.60, is less than $\frac{2}{3}$ .

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Question

$\frac{2}{7}\cdot\frac{1}{4}=?$

What is the result of the above equation?

Answer

So we first multiply the numerators:

$2\cdot1=2$

We then multiply the denominators

$7\cdot4=28$

Our resulting fraction is:

$\frac{2}{28}$

We reduce by dividing by 2 from the numerator and denominator.

For the numerator:

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

For the denominator:

$\frac{28}{2}=14$

The resulting fraction is:

$\frac{1}{14}$

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Question

$\frac{3}{4}\cdot\frac{4}{6}=?$

What is the result of the above equation?

Answer

We first multiply the numerators:

$3 \cdot4=12$

Then we multiply the denominators:

$4\cdot6=24$

The resulting fraction is:

$\frac{12}{24}$

We divide the numerator and denominator by 12.

The numerator is:

$\frac{12}{12}=1$

The denominator results in:

$\frac{24}{12}=2$

Our resulting fraction is:

$\frac{1}{2}$

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Question

What is the result of the below equation?

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

Answer

First, multiply the terms in the numerators:

$2\times3=6$

Then multiply the terms in the denominators:

$5\times6=30$

This results in the fraction $\frac{6}{30}$ .

Notice that both the numerator and denominator have a common factor of 6. Dividing both the numerator and denominator by 6 results in the simplest form fraction $\frac{1}{5}$ .

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Question

What is the result of the below equation?

$\frac{9}{10}\times \frac{2}{3}=?$

Answer

First, multiply the numerators of the terms:

$9\times2=18$

Then, multiply the denominators of the terms:

$10\times3=30$

The resulting fraction is $\frac{18}{30}$ . Note that the numerator and denominator both have a common factor of 6. Dividing both parts of the fraction by 6, we get the simplest form fraction $\frac{3}{5}$ .

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Question

What is the result of the below equation?

$\frac{7}{12}\times\frac{4}{5}=?$

Answer

First, multiply the numerators of each term:

$7\times4=28$

Then, multiply the denominators:

$12\times5=60$

This results in a fraction of $\frac{28}{60}$ . Since the numerator and denominator both have a common factor of 4, reduce both the numerator and denominator by dividing by 4 to get the simplest form fraction $\frac{7}{15}$ .

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Question

What is the reciprocal of $\frac{4}{7}$ ?

Answer

To get the reciprocal of a fraction, you simply switch the numerator and the denominator.

In our case our numerator is and our denominator is .

So $\frac{4}{7}$ becomes $\frac{7}{4}$ .

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Question

What is the reciprocal of $\small 8$ multiplied by the reciprocal of $\small \small \frac{3}{2}$ ?

Answer

To find the reciprocal of a fraction, we simply need to switch the numerator and the denominator: for example, the reciprocal of a fraction $\small \frac{a}{b}$ is $\small \frac{b}{a}$ .

With integers, it helps to remember that all integers are really fractions with a denominator of $\small 1$ :

$\small 4=\frac{4}{1}$ , $\small 2=\frac{2}{1}$ , and $\small 17=\frac{17}{1}$

The reciprocals of these numbers are $\small \frac{1}{4}, \frac{1}{2},$ and $\small \frac{1}{17}$ respectively.

Therefore, to solve the problem, we first need to find the reciprocals of $\small 8$ and $\small \frac{3}{2}$ . If we keep in mind that $\small 8=\frac{8}{1}$ , we can determine that the reciprocals are $\small \frac{1}{8}$ and $\small \frac{2}{3}$ , respectively. The product of these two numbers is:

$\small \frac{1}{8}\times\frac{2}{3}=\frac{1\times2}{8\times3}=\frac{2}{24}=\frac{1}{12}$

$\small \frac{1}{12}$ is our final answer.

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Question

What is the sum of the reciprocal of $\frac{1}{4}$ and ?

Answer

To find the reciprocal of a fraction, flip the numerator and the denominator.

Thus, the reciprocal of $\frac{1}{4}$ is $\frac{4}{1}=4$ .

Then we need to find the sum of 4 and 7, which is 11.

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Question

Compute: $\frac{x}{\left(2x)^{-2}\right }$

Answer

We will need to rewrite this in order to eliminate the negative exponent in the problem.

Because the denominator has a negative exponent, that is the same as having a positive exponent in the numerator. Therefore we can rewrite the problem as follows and then multiply.

$\frac{x}{\left(2x)^{-2}\right } = x\cdot (2x)^2 = x\cdot4x^2 = 4x^3$

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Question

Evaluate:

$\frac{3}{7} \div \frac{7}{3}$

Answer

To divide a term by a fraction, take the reciprocal of the fraction.

Then mutiply both terms.

$\frac{3}{7} \div \frac{7}{3}= \frac{3}{7} \cdot \frac{3}{7} = \frac{9}{49}$

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