Subtracting Mixed Numbers
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:
- Convert each mixed number to improper fraction.
- Subtract improper fractions (using subtraction of fractions with unlike denominators)
- Convert improper fraction to mixed number if needed (and if possible).
Example 1. Find $$${1}\frac{{3}}{{5}}-{2}\frac{{4}}{{9}}$$$.
Convert each mixed number into improper fraction: $$${1}\frac{{3}}{{5}}=\frac{{8}}{{5}}$$$ and $$${2}\frac{{4}}{{9}}=\frac{{22}}{{9}}$$$.
Now subtract fractions: $$$\frac{{8}}{{5}}-\frac{{22}}{{9}}=\frac{{{8}\cdot{9}}}{{{5}\cdot{9}}}-\frac{{{22}\cdot{5}}}{{{9}\cdot{5}}}=\frac{{72}}{{45}}-\frac{{110}}{{45}}=-\frac{{38}}{{45}}$$$.
Convert fraction to mixed number: can't convert because fraction is proper.
Answer: $$$-\frac{{38}}{{45}}$$$.
Next example.
Example 2. Find $$$-{2}\frac{{1}}{{6}}-{3}\frac{{1}}{{2}}$$$.
Convert each mixed number into improper fraction: $$$-{2}\frac{{1}}{{6}}=-\frac{{13}}{{6}}$$$ and $$${3}\frac{{1}}{{2}}=\frac{{7}}{{2}}$$$.
Now subtract fractions: $$$-\frac{{13}}{{6}}-\frac{{7}}{{2}}=-\frac{{13}}{{6}}-\frac{{{7}\cdot{3}}}{{{2}\cdot{3}}}=-\frac{{13}}{{6}}-\frac{{21}}{{6}}=-\frac{{34}}{{6}}$$$.
Reduce fraction: $$$-\frac{{34}}{{6}}=-\frac{{17}}{{3}}$$$.
Convert fraction to mixed number: $$$-\frac{{17}}{{3}}=-{5}\frac{{2}}{{3}}$$$.
Answer: $$$-\frac{{17}}{{3}}=-{5}\frac{{2}}{{3}}$$$.
Next example.
Example 3. Find $$$-{2}-{5}\frac{{8}}{{11}}$$$.
Since first number has no fractional part, we can easier subtract numbers.
Subtract integer parts: $$$-{2}-{5}=-{7}$$$ 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 $$${2}-{\left(-{5}\frac{{8}}{{11}}\right)}={7}\frac{{8}}{{11}}$$$).
Answer: $$$-{7}\frac{{8}}{{11}}=-\frac{{85}}{{11}}$$$.
Now, take pen and paper and do following exercises.
Exercise 1. Find $$${5}\frac{{1}}{{6}}-{3}\frac{{7}}{{8}}$$$.
Answer: $$$\frac{{31}}{{24}}={1}\frac{{7}}{{24}}$$$.
Next exercise.
Exercise 2. Find $$${7}\frac{{4}}{{9}}-{\left(-{2}\frac{{5}}{{6}}\right)}$$$.
Answer: $$$\frac{{185}}{{18}}={10}\frac{{5}}{{18}}$$$.
Next exercise.
Exercise 3. Find $$$-{5}\frac{{1}}{{6}}-{\left(-{3}\frac{{7}}{{8}}\right)}$$$.
Answer: $$$-\frac{{31}}{{24}}=-{1}\frac{{7}}{{24}}$$$.
Next exercise.
Exercise 4. Find $$${3}\frac{{5}}{{6}}-\frac{{5}}{{7}}$$$.
$$${3}\frac{{5}}{{6}}-\frac{{5}}{{7}}=\frac{{23}}{{6}}-\frac{{5}}{{7}}=\frac{{161}}{{42}}-\frac{{30}}{{42}}=\frac{{131}}{{42}}={3}\frac{{5}}{{42}}$$$.
Answer: $$$\frac{{131}}{{42}}={3}\frac{{5}}{{42}}$$$.
Next exercise.
Exercise 5. Find $$${5}\frac{{1}}{{6}}-{7}$$$.
Here we just can't add fractional parts to obtain $$$-{2}\frac{{1}}{{6}}$$$. This is not correct, because both numbers are positive.
We do it as always.
$$${5}\frac{{1}}{{6}}-{7}=\frac{{31}}{{6}}-\frac{{42}}{{6}}=-\frac{{11}}{{6}}=-{1}\frac{{5}}{{6}}$$$.
Answer: $$$-\frac{{11}}{{6}}=-{1}\frac{{5}}{{6}}$$$.
If you are not sure whether it is possible to subtract integer parts, use the three-step method. It guarantees correct answer.