Simplify each of the numerical questions.

Question

Q1) $-8 \frac{1}{4}+(-4 \frac{5}{8})-(-6 \frac{3}{8})$

Q2) $9 \frac{1}{3}-12 \frac{1}{2}+(-4 \frac{1}{6})-(-1 \frac{1}{6})$

Answer 1

For the record, most people agree that "compound numbers" are a terrible invention, and a number such as $8\frac{1}{4}$ should instead be written as $8+\frac{1}{4}$. America is the only country that writes numbers this way to my knowledge, and this often causes confusion. That being said, I recommend breaking each compound number as the sum of a whole number and a fraction, keeping in mind the negative signs, and combining numbers as best you can.

Keep in mind the following rules:

• You may always multiply a number by "one" and it will stay the same. The trick is what way of writing "one" you use. In this context, you can multiply by something like $\frac{2}{2}$ since $\frac{2}{2}=1$. You can use this to change fractions of one denominator into fractions of another denominator. For example, changing $\frac{3}{4}$ into $\frac{6}{8}$ since $\frac{3}{4}=\frac{3}{4}\cdot 1 = \frac{3}{4}\cdot \frac{2}{2} = \frac{3\cdot 2}{4\cdot 2}=\frac{6}{8}$
• If you are subtracting something, it is the same as "adding negative one times that something". E.g. $5-(2+1) = 5+(-1)(2+1)$
• If you are multiplying a parenthesis containing numbers being added, it is the same as multiplying each number inside by that. $(-1)(2+1) = (-1)(2)+(-1)(1)=(-2)+(-1)=-2-1$
• If two fractions have the same denominator, you can combine them: E.g. $\frac{1}{4}+\frac{2}{4} = \frac{1+2}{4}=\frac{3}{4}$

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Example problem:

$\begin{array}{ll} 1\frac{1}{2}+3\frac{1}{3}-5\frac{1}{3} &= 1+\frac{1}{2}+3+\frac{1}{3}+(-1)(5+\frac{1}{3})\\ & = 1+\frac{1}{2}+3+\frac{1}{3}-5-\frac{1}{3}\\ & = 1 + \frac{3}{6}+3+\frac{2}{6}-5-\frac{2}{6}\\ & = 1+3-5 + \frac{3+2-2}{6}\\ & = -1+\frac{3}{6}\\ & = -1 + \frac{1}{2}\\ & = \frac{-2}{2} + \frac{1}{2}\\ & = \frac{-2+1}{2}\\ & = \frac{-1}{2}\end{array}$

As you become better with arithmetic, you can do multiple steps in your head without thinking too hard about it.