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Subtracting Mixed Numbers

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Subtracting mixed numbers is quite easy.

We know that mixed number consists of integer part and fractional part.

To subtract mixed numbers three steps are needed:

  1. Convert each mixed number to improper fraction.
  2. Subtract improper fractions (using subtraction of fractions with unlike denominators)
  3. Convert improper fraction to mixed number if needed (and if possible).

Example 1. Find 135−249{1}\frac{{3}}{{5}}-{2}\frac{{4}}{{9}}.

Convert each mixed number into improper fraction: 135=85{1}\frac{{3}}{{5}}=\frac{{8}}{{5}} and 249=229{2}\frac{{4}}{{9}}=\frac{{22}}{{9}}.

Now subtract fractions: 85−229=8⋅95⋅9−22⋅59⋅5=7245−11045=−3845\frac{{8}}{{5}}-\frac{{22}}{{9}}=\frac{{{8}\cdot{9}}}{{{5}\cdot{9}}}-\frac{{{22}\cdot{5}}}{{{9}\cdot{5}}}=\frac{{72}}{{45}}-\frac{{110}}{{45}}=-\frac{{38}}{{45}}.

Convert fraction to mixed number: can't convert because fraction is proper.

Answer: −3845-\frac{{38}}{{45}}.

Next example.

Example 2. Find −216−312-{2}\frac{{1}}{{6}}-{3}\frac{{1}}{{2}}.

Convert each mixed number into improper fraction: −216=−136-{2}\frac{{1}}{{6}}=-\frac{{13}}{{6}} and 312=72{3}\frac{{1}}{{2}}=\frac{{7}}{{2}}.

Now subtract fractions: −136−72=−136−7⋅32⋅3=−136−216=−346-\frac{{13}}{{6}}-\frac{{7}}{{2}}=-\frac{{13}}{{6}}-\frac{{{7}\cdot{3}}}{{{2}\cdot{3}}}=-\frac{{13}}{{6}}-\frac{{21}}{{6}}=-\frac{{34}}{{6}}.

Reduce fraction: −346=−173-\frac{{34}}{{6}}=-\frac{{17}}{{3}}.

Convert fraction to mixed number: −173=−523-\frac{{17}}{{3}}=-{5}\frac{{2}}{{3}}.

Answer: −173=−523-\frac{{17}}{{3}}=-{5}\frac{{2}}{{3}}.

Next example.

Example 3. Find −2−5811-{2}-{5}\frac{{8}}{{11}}.

Since first number has no fractional part, we can easier subtract numbers.

Subtract integer parts: −2−5=−7-{2}-{5}=-{7} and fractional part leave the same.

Warning. This works only when we subtract either negative and positive numbers (this case) or positive and negative numbers (like 2−(−5811)=7811{2}-{\left(-{5}\frac{{8}}{{11}}\right)}={7}\frac{{8}}{{11}}).

Answer: −7811=−8511-{7}\frac{{8}}{{11}}=-\frac{{85}}{{11}}.

Now, take pen and paper and do following exercises.

Exercise 1. Find 516−378{5}\frac{{1}}{{6}}-{3}\frac{{7}}{{8}}.

Answer: 3124=1724\frac{{31}}{{24}}={1}\frac{{7}}{{24}}.

Next exercise.

Exercise 2. Find 749−(−256){7}\frac{{4}}{{9}}-{\left(-{2}\frac{{5}}{{6}}\right)}.

Answer: 18518=10518\frac{{185}}{{18}}={10}\frac{{5}}{{18}}.

Next exercise.

Exercise 3. Find −516−(−378)-{5}\frac{{1}}{{6}}-{\left(-{3}\frac{{7}}{{8}}\right)}.

Answer: −3124=−1724-\frac{{31}}{{24}}=-{1}\frac{{7}}{{24}}.

Next exercise.

Exercise 4. Find 356−57{3}\frac{{5}}{{6}}-\frac{{5}}{{7}}.

356−57=236−57=16142−3042=13142=3542{3}\frac{{5}}{{6}}-\frac{{5}}{{7}}=\frac{{23}}{{6}}-\frac{{5}}{{7}}=\frac{{161}}{{42}}-\frac{{30}}{{42}}=\frac{{131}}{{42}}={3}\frac{{5}}{{42}}.

Answer: 13142=3542\frac{{131}}{{42}}={3}\frac{{5}}{{42}}.

Next exercise.

Exercise 5. Find 516−7{5}\frac{{1}}{{6}}-{7}.

Here we just can't add fractional parts to obtain −216-{2}\frac{{1}}{{6}}. This is not correct, because both numbers are positive.

We do it as always.

516−7=316−426=−116=−156{5}\frac{{1}}{{6}}-{7}=\frac{{31}}{{6}}-\frac{{42}}{{6}}=-\frac{{11}}{{6}}=-{1}\frac{{5}}{{6}}.

Answer: −116=−156-\frac{{11}}{{6}}=-{1}\frac{{5}}{{6}}.

If you are not sure whether it is possible to subtract integer parts, use the three-step method. It guarantees correct answer.