Operations with Fractions - Basic Arithmetic

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Question

Simplify $\frac{2}{3}+\frac{1}{4}+\frac{5}{6}$

Answer

Find the least common denominator (LCD) and convert each fraction to the LCD and then add. Simplify as necessary.

$\frac{2}{3}+\frac{1}{4}+\frac{5}{6}=\frac{8}{12}+\frac{3}{12}+\frac{10}{12}=\frac{21}{12}$

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

$\frac{21}{12}=\frac{7}{4}=1\frac{3}{4}$

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Question

Solve:

$\frac{3}{2}+\frac{7}{9}=$

Answer

The least common multiple of 2 and 9 is 18.

$\frac{3}{2}=\frac{27}{18}$

$\frac{7}{9}=\frac{14}{18}$

$\frac{27}{18}+\frac{14}{18}=\frac{27+14}{18}=\frac{41}{18}$

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Question

Simplify the following fraction:

$\frac{12}{39}$

Answer

Determine the factors of both the numerator and the denominator:

$12 = 2 \times 2 \times 3$

$39 = 3 \times 13$

We notice that 3 is a factor of both 12 and 39 so we can simplify by dividing both 12 and 39 by 3.

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

$\frac{39}{3}=13$

The result is therefore,

$\frac{4}{13}$

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Question

Find the simplified result:

$\frac{5}{6} + \frac{7}{8}$

Answer

Start by making both fractions into the same denominator. One option is $6 \times 8 = 48$

Then adjust the numerators by multiplying each fraction's numerator by the other fraction's denominator:

$\frac{5}{6}\cdot \frac{8}{8}+ \frac{7}{8}\cdot \frac{6}{6}$

$=\frac{40}{48}+\frac{42}{48}$

Then add the adjusted numerators:

$=\frac{82}{48}$

Then we simplify by dividing both numerator and denominator by 2:

$=\frac{2 \cdot 41}{2 \cdot 24}=\frac{41}{24}$

which gives us the final result.

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Question

Solve the following expression:

$\frac{1}{2}+\frac{1}{6}$

Answer

The correct answer is $\frac{2}{3}$ .

In order to start solving this problem, we need to get each term to have a common denominator.

The common denominator between 2 and 6 is 6. We can rewrite the expression and solve using the following method:

$\frac{1}{2}+\frac{1}{6} = \frac{3}{6}+\frac{1}{6} = \frac{4}{6} = \frac{2}{3}$

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Question

Please read the following question thoroughly, and then choose the answer which is most correct.

If Billy has of a gallon of water, and Bobby has of a gallon of water, how much water do they have between the two of them?

Answer

To complete this question, you need to find the least common denominator in order to add the fractions. Cross multiply the fractions so that they are in the same terms, and then add. To do this, simply multiply each fraction by the denominator of the other, over itself (so that the fraction equals 1). Follow the steps below:

$\frac{2}{3}+ \frac{1}{8} = ?$

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

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

$\frac{16}{24} + \frac{3}{24} = \frac{19}{24}$

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Question

Evaluate:

$-\frac{6}{5}+\frac{5}{6}$

Answer

In fractions, to add or subtract the numerator, the denominators must be equal.

Find the least common denominator, multiply the top and bottom with what was multiplied on the denominator, and simplify.

$-\frac{6}{5}+\frac{5}{6} = -\frac{6\cdot6}{5\cdot6}+\frac{5\cdot5}{6\cdot5}=-\frac{36}{30}+\frac{25}{30}$

= $-\frac{11}{30}$

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Question

Determine the answer as a mixed number:

$\frac{3}{4} + \frac{5}{6} =$

Answer

1. Find the least common denominator:

The lowest number that both 4 and 6 can go into is 12 meaning that it is the least common denominator.

2. Use the new least common denominator to create two equivalent fractions with a denominator of 12.

$(\frac{3}{4})(\frac{3}{3})=\frac{9}{12}$

$(\frac{5}{6})(\frac{2}{2})=\frac{10}{12}$

3. Add the two fractions together.

$\frac{9}{12}+\frac{10}{12}=\frac{19}{12}$

4. Convert the fraction to a mixed number:

$\frac{19}{12}=1 \frac{7}{12}$

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Question

What is the sum of $\small \small \frac{4}{9}$ and $\small \frac{2}{5}$ ?

Answer

When adding two fractions, the denominators must be the same. When the denominators are the same, we merely add the numerators together to see what the sum is. The trickiest part is transforming the two fractions into fractions with the same denominator.

When we transform a fraction, we are multiplying it by $\small 1$ . For example, $\small \frac{2}{3}$ is the same fraction as $\small \frac{4}{6}$ because $\small \frac{2}{3}\times\frac{2}{2}=\frac{2\times 2}{3\times 2}=\frac{4}{6}$ . Because $\small \frac{2}{2}=1$ , we are not changing the value of the fraction, we are merely expressing the same value in a different form. This is essential when adding fractions with different denominators.

To rewrite these fractions with the same denominator, we need to find the Least Common Denominator (LCD). In this case, the least common denominator is $\small 9\times5=45$ , as $\small 45$ contains both $\small 9$ and $\small 5$ as factors. Our next step is to rewrite our two fractions as fractions with denominators of $\small 45$ .

To transform our first fraction, $\small \frac{4}{9}$ , into a fraction with $\small 45$ as its denominator, we need to multiply the fraction by $\small \frac{5}{5}$ .

To transform our second fraction, $\small \frac{2}{5}$ , into a fraction with as its denominator, we need to multiply the fraction by $\small \frac{9}{9}$ . Therefore, our answer becomes:

$\small \frac{4}{9}+\frac{2}{5}=\frac{4}{9}\times\frac{5}{5}+\frac{2}{5}\times\frac{9}{9}$

$\small \frac{4\times5}{9\times5}+\frac{2\times9}{5\times9}=\frac{20}{45}+\frac{18}{45}$

$\small \frac{20+18}{45}=\frac{38}{45}$

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Question

Add these fractions:

$\frac{3}{5}+\frac{2}{7}$

Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 7 over 7 and the second fraction by 5 over 5.

$\frac{3}{5}\frac{7}{7}=\frac{21}{35}$

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

Add these fractions together, to get the final answer.

$\frac{21}{35}+\frac{10}{35}=\frac{31}{35}$

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Question

Add these fractions:

$\frac{1}{7}+\frac{5}{6}$

Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 6 over 6 and the second fraction by 7 over 7.

$\frac{1}{7}\frac{6}{6}=\frac{6}{42}$

$\frac{5}{6}\frac{7}{7}=\frac{35}{42}$

Add the numerators of the fractions to get the final answer.

$\frac{6}{42}+\frac{35}{42}=\frac{41}{42}$

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Question

Add these fractions:

$\frac{3}{7}+\frac{2}{5}$

Answer

To solve this we need to first find common denominators. We do that by multiplying the first fraction by 5 over 5 and the second fraction by 7 over 7.

$\frac{3}{7}\frac{5}{5}=\frac{15}{35}$

$\frac{2}{5}\frac{7}{7}=\frac{14}{35}$

Add the numerators of the fractions to get the final answer.

$\frac{15}{35}+\frac{14}{35}=\frac{29}{35}$

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Question

Simplify $\frac{5}{3}\div\frac{7}{6}$ .

Answer

To solve, we must turn the division problem into a multiplication problem by "flipping" the second fraction (dividing by a fraction is the same as multiplying by its reciprocal):

$\frac{5}{3}\div\frac{7}{6}=\frac{5}{3}\cdot\frac{6}{7}$ .

Then, we multiply the numerators followed by the denomenators:

$\frac{5\cdot6}{3\cdot7}=\frac{30}{21}$ .

Lastly, the fraction must be simplified by a factor of 3:

$\frac{30}{21}=\frac{10}{7}$ , which gives us our final answer.

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Question

Solve the following:

$\frac{4}{5}\div4$

Answer

The correct answer is $\frac{1}{5}$ and can be found by using reciprocals as shown below:

$\frac{4}{5}\div\frac{4}{1} = \frac{4}{5}\ast \frac{1}{4} = \frac{4}{20}=\frac{1}{5}$

The first step is to identify that 4 can be written as 4 divided by 1. You can then take the first fraction and multiple it by the reciprocal of the second fraction, which can be found by switching the numerator and denominator. From there, you solve and simplify until you get to the final answer.

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Question

Please choose the best answer for the question below.

What is divided by ?

Answer

To divide fractions, simply invert the second fraction and multiply. So, divided by becomes:

$6/7 \cdot 5/2 = 30/14$

reduce to find the correct answer:

.

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Question

Determine the answer as an improper fraction:

$\frac{16}{8} \div \frac{3}{4}=$

Answer

1. Invert the second fraction:

$\frac{3}{4}\rightarrow \frac{4}{3}$

2. Multiply the two fractions:

$\frac{16}{8}\times \frac{4}{3}=\frac{64}{24}$

3. Reduce the improper fraction:

$\frac{64}{24}=\frac{8}{8}\times \frac{8}{3}$

So the improper fraction can be reduced to $\frac{8}{3}$ .

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Question

Evaluate the following division problem and express the resulting fraction in its simplest form.

$\frac{16}{3}\div \frac{5}{12}$

Answer

In order to divide two fractions you simply find the inverse of either fraction, and multiply that by the other fraction.

To find the inverse of a fraction, you just switch the numerator and the denominator.

The inverse of $\frac{5}{12}$ is $\frac{12}{5}$ .

So to divide:

$\frac{16}{3}\div \frac{5}{12}$

We turn it into the following multiplication problem:

$\frac{16}{3}\times\frac{12}{5}$

We multiply the numerators by each other and the denominators by each other to carry out the multiplication:

$\frac{16\times 12}{3\times 5}=\frac{192}{15}$

Both 192 and 15 are divisible by 3, so we divide 192 by 3 and 15 by 3 and are left with:

$\frac{64}{5}$

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Question

Evaluate: $\frac{7}{6}\div 2\frac{3}{8}$

Answer

Change all the fractions to improper fractions.

$\frac{7}{6}\div 2\frac{3}{8}=\frac{7}{6}\div \frac{19}{8}$

Convert the division sign to a multiplication sign, but flip the second term. Simplify.

$\frac{7}{6}\times \frac{8}{19}= \frac{7}{3}\times \frac{4}{19}= \frac{28}{57}$

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Question

Divide and simplify:

$\frac{2}{11}\div \frac{9}{22}$ .

Answer

Multiplying fractions is a two-step process. First, you must flip the second fraction (make it its reciprocal) and then set up the equation as a multiplication problem:

$\frac{2}{11}\cdot \frac{22}{}9$ .

Then, cross-reduce: 11 goes into both 11 and 22, so you can take that out so it looks like:

$\frac{2}{1}\cdot \frac{2}{9}$ .

Then, multiply straight across so that you get $\frac{4}{}9$ .

You can't simplify any further, so that's your answer!

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Question

Divide these fractions:

$\frac{15}{30}\div15$

Answer

When dividing fractions, first we need to flip the second fraction. Then multiply the numerators together and multiply the denominators together:

$\frac{15}{30}\div15=\frac{15}{30}\frac{1}{15}=\frac{151}{3015}$

Simplify the fraction to get the final answer:

$\frac{151}{30*15}=\frac{1}{30}$

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