Adding And Subtracting Fractions

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 8 and 4 is 8.

Therefore we only need to use equivalent fractions to change the denominator of the second fraction. We do this by multiplying the numerator and denominator by 2, because 4\times2=8.

\cfrac{1}{4}\times\cfrac{2}{2}=\cfrac{1\times{2}}{4\times{2}}=\cfrac{2}{8}

We now have the two fractions \cfrac{3}{8}+\cfrac{2}{8} both with a common denominator of 8.

5 and 8 do not share any common factors except 1 and so the fraction is already in its simplest form.

The solution is \cfrac{5}{8}.

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 2 and 11 is 22.

So we multiply the numerator and denominator of the first fraction by 11 to achieve a denominator of 22.

\cfrac{1}{2}\times\cfrac{11}{11}=\cfrac{1\times{11}}{2\times{11}}=\cfrac{11}{22}

And, we multiply the numerator and denominator of the second fraction by 2 to achieve a denominator of 22.

\cfrac{5}{11}\times\cfrac{2}{2}=\cfrac{5\times{2}}{11\times{2}}=\cfrac{10}{22}

We now have the two fractions \cfrac{11}{22}+\cfrac{10}{22} both with a common denominator of 22.

21 and 22 do not share any common factors except 1 and so the fraction is already in its simplest form.

The solution is \cfrac{21}{22}.

The fractions have a common denominator of 12.

2 and 12 share a common factor of 2. Therefore we can divide the numerator and denominator by 2 in order to simplify the fraction.

\cfrac{2}{12}=\cfrac{2\div 2}{12\div 2}=\cfrac{1}{6}

The solution is \cfrac{1}{6}.

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 15 and 5 is 15.

Therefore we only need to use equivalent fractions for changing the denominator of the second fraction. We do this by multiplying the numerator and denominator by 3, because 5\times3=15.

\cfrac{2}{5}\times\cfrac{3}{3}=\cfrac{2\times{3}}{5\times{3}}=\cfrac{6}{15}

We now have the two fractions \cfrac{11}{15}-\cfrac{6}{15} both with a common denominator of 15.

5 and 15 share a common factor of 5. Therefore we can divide the numerator and denominator by 5 in order to simplify the fraction.

\cfrac{5}{15}=\cfrac{5\div 5}{15\div 5}=\cfrac{1}{3}

The solution is \cfrac{1}{3}.

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 6 and 9 is 18.

So we multiply the numerator and denominator of the first fraction by 3 to achieve a denominator of 18, because 6\times 3=18.

\cfrac{5}{6}\times\cfrac{3}{3}=\cfrac{5\times{3}}{6\times{3}}=\cfrac{15}{18}

And, we multiply the numerator and denominator of the second fraction by 2 to achieve a denominator of 18 because 9\times 2=18.

\cfrac{7}{9}\times\cfrac{2}{2}=\cfrac{7\times{2}}{9\times{2}}=\cfrac{14}{18}

We now have the two fractions \cfrac{15}{18}-\cfrac{14}{18} both with a common denominator of 18.

1 and 18 do not share any common factors except 1 and so the fraction is already in its simplest form.

The solution is \cfrac{1}{18}.

2\cfrac{5}{6}=\cfrac{(6\times{2})+5}{6}=\cfrac{12+5}{6}=\cfrac{17}{6}

We now have the two fractions \cfrac{7}{4}+\cfrac{17}{6}.

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 4 and 6 is 12.

So we multiply the numerator and denominator of the first fraction by 3 to achieve a denominator of 12 because 4\times 3=12.

\cfrac{7}{4}\times\cfrac{3}{3}=\cfrac{7\times{3}}{4\times{3}}=\cfrac{21}{12}

We multiply the numerator and denominator of the second fraction by 2 to achieve a denominator of 12 because 6\times 2=12.

\cfrac{17}{6}\times\cfrac{2}{2}=\cfrac{17\times{2}}{6\times{2}}=\cfrac{34}{12}

We now have the two fractions \cfrac{21}{12}+\cfrac{34}{12} both with a common denominator of 12.

55 and 12 do not share any common factors except 1 and so the fraction is already in its simplest form. However, as \cfrac{55}{12} is an improper fraction, we should convert this back to a mixed number for the solution.

55\div 12 = 4 \ (remainder \ 7)

Therefore the mixed number has 4 as a whole number, and 7 remains to be divided by 12, giving 4\cfrac{7}{12}.

The solution is 4\cfrac{7}{12}.

1\cfrac{2}{3}=\cfrac{(3\times{1})+2}{3}=\cfrac{3+2}{3}=\cfrac{5}{3}

We now have the two fractions \cfrac{7}{2}-\cfrac{5}{3}.

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 2 and 3 is 6.

So we multiply the numerator and denominator of the first fraction by 3 to achieve a denominator of 6.

\cfrac{7}{2}\times\cfrac{3}{3}=\cfrac{7\times{3}}{2\times{3}}=\cfrac{21}{6}

We multiply the numerator and denominator of the second fraction by 2 to achieve a denominator of 6.

\cfrac{5}{3}\times\cfrac{2}{2}=\cfrac{5\times{2}}{3\times{2}}=\cfrac{10}{6}

We now have the two fractions \cfrac{21}{6}-\cfrac{10}{6} both with a common denominator of 6.

11 and 6 do not share any common factors except 1 and so the fraction is already in its simplest form.

As \cfrac{11}{6} is an improper fraction, we should convert this back to a mixed number for the solution.

11\div 6 = 1 \ (remainder \ 5)

Therefore the mixed number has 1 as a whole number, and 5 remains to be divided by 6, giving 1\cfrac{5}{6}.

The solution is 1\cfrac{5}{6}.

2\cfrac{4}{5}=\cfrac{(5\times{2})+4}{5}=\cfrac{10+4}{5}=\cfrac{14}{5}

1\cfrac{5}{6}=\cfrac{(6\times{1})+5}{6}=\cfrac{6+5}{6}=\cfrac{11}{6}

We now have the three fractions \cfrac{14}{3}+\cfrac{14}{5}-\cfrac{11}{6}.

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 3,5 and 6 is 30.

We therefore need to determine an equivalent fraction for each of the three fractions with a denominator of 30.

We need to multiply the numerator and denominator of the first fraction by 10 to achieve a denominator of 30.

\cfrac{14}{3}\times\cfrac{10}{10}=\cfrac{14\times{10}}{3\times{10}}=\cfrac{140}{30}

We need to multiply the numerator and denominator of the second fraction by 6 to achieve a denominator of 30.

\cfrac{14}{5}\times\cfrac{6}{6}=\cfrac{14\times{6}}{5\times{6}}=\cfrac{84}{30}

We need to multiply the numerator and denominator of the third fraction by 5 to achieve a denominator of 30.

\cfrac{11}{6}\times\cfrac{5}{5}=\cfrac{11\times{5}}{6\times{5}}=\cfrac{55}{30}

We now have the three fractions \cfrac{140}{30}+\cfrac{84}{30}-\cfrac{55}{30} all with a common denominator of 30.

169 and 30 do not share any common factors except 1 and so the fraction is already in its simplest form.

As \cfrac{169}{30} is an improper fraction, we should convert this back to a mixed number for the solution.

169\div 30 = 5 \ (remainder \ 19)

Therefore the mixed number has 5 as a whole number, and 19 remains to be divided by 30, giving 5\cfrac{19}{30}.

The solution is 5\cfrac{19}{30}.

5+3-4+\cfrac{2}{7}+\cfrac{4}{7}-\cfrac{5}{7}

The fractions all have the common denominator of 7.

The whole number answer is 4.

The fraction answer is \cfrac{1}{7}.

4+\cfrac{1}{7}=4\cfrac{1}{7}

1 and 7 do not share any common factors except 1 and so the fraction is already in its simplest form.

The solution is 4\cfrac{1}{7}.

1+6+\cfrac{2}{3}+\cfrac{10}{21}

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 3 and 21 is 21.

Therefore we only need to use equivalent fractions for changing the denominator of the first fraction. We do this by multiplying the numerator and denominator by 7, because 3\times7=21.

\cfrac{2}{3}=\cfrac{2}{3}\times\cfrac{7}{7}=\cfrac{2\times 7}{3\times 7}=\cfrac{14}{21}

The whole number answer is 7.

The fraction answer is \cfrac{24}{21}.

This fraction can be simplified and changed to a mixed number first.

\cfrac{24}{21}=\cfrac{24\div 3}{21\div 3}=\cfrac{8}{7}

8\div 7 = 1 \ (remainder \ 1)

Therefore the mixed number has 1 as a whole number, and 1 remains to be divided by 7, giving 1\cfrac{1}{7}.

Putting the answers together,

7+1\cfrac{1}{7}=8\cfrac{1}{7}.

The solution is 8\cfrac{1}{7}.

5-2+\cfrac{1}{10}-\cfrac{7}{15}

These fractions have different denominators, therefore we need to use equivalent fractions to write them with a common denominator. The common denominator we should use will be the lowest common multiple (LCM) of the denominators.

The lowest common multiple of 10 and 15 is 30.

So, we multiply the numerator and denominator of the first fraction by 3, because 10\times3=30.

\cfrac{1}{10}=\cfrac{1}{10}\times\cfrac{3}{3}=\cfrac{1\times 3}{10\times 3}=\cfrac{3}{30}

Next, we multiply the numerator and denominator of the second fraction by 2, because 15\times2=30.

\cfrac{7}{15}=\cfrac{7}{15}\times\cfrac{2}{2}=\cfrac{7\times 2}{15\times 2}=\cfrac{14}{30}

The whole number answer is 3.

The fraction answer is -\cfrac{11}{30}.

Putting the answers together,

3-\cfrac{11}{30}.

Here, we can take one from the whole number and write it as a fraction with a denominator of 30 and then we can finish the calculation.

3-\cfrac{11}{30}=2+1-\cfrac{11}{30}=2+\cfrac{30}{30}-\cfrac{11}{30}=2+\cfrac{30-11}{30}=2+\cfrac{19}{30}=2\cfrac{19}{30}

19 and 30 do not share any common factors except 1 and so the fraction is already in its simplest form.

The solution is 2\cfrac{19}{30}.