1.03 Multiply and divide rational numbers
Multiply and divide rational numbers
Just like with adding and subtracting different types of rational numbers, it is helpful to convert numbers to the same form when multiplying or dividing them.
- Multiplying decimals: multiply the numbers as if there were no decimal points at all. Then, count the numbers after the decimal points in the original numbers, and place the decimal in the total number of places to the left.
Multiply 4.83 by 5.7
\begin{array}{c} & & & &4&8&3 \\ &\cdot & & & &5&7 \\ \hline & & &3&3&8&1 \\ &+ &2&4&1&5&0 \\ \hline & &2&7&5&3&1 \end{array}
In this case, the original numbers are 4.83, which has two decimal places, and 5.7, which has one decimal place. So their product will have 2+1=3 decimal places.
- Dividing decimals: use the fact that multiplying by 10 moves the decimal one place to the right. Take the divisor and multiply it by 10 until it is a whole number, then multiply the dividend by that same power of 10. We can then use long division rules for whole numbers.
Divide 5.6 by 0.7
| \displaystyle 5.6 \div 0.7 | \displaystyle = | \displaystyle (5.6 \cdot 10) \div (0.7 \cdot 10) | Convert divisor to a whole number by multiplying by both values by 10 |
| \displaystyle = | \displaystyle 56 \div 7 | Divide 56 by 7 using long division. | |
| \displaystyle = | \displaystyle 8 | Evaluate |
Multiplying fractions: multiply the numerators, multiply the denominators, and simplify.
Multiply \dfrac{2}{3} by \dfrac{5}{4}
| \displaystyle \dfrac{2}{3} \cdot \dfrac{5}{4} | \displaystyle = | \displaystyle \dfrac{2 \cdot 5}{3 \cdot 4} | Multiply the numerators |
| \displaystyle = | \displaystyle \dfrac{10}{3 \cdot 4} | Multiply the denominators | |
| \displaystyle = | \displaystyle \dfrac{10}{12} | Simplify the fraction | |
| \displaystyle = | \displaystyle \dfrac{5}{6} |
Dividing fractions: take the reciprocal of the divisor and multiply. Then, follow the rules for multiplication.
Divide \dfrac{4}{5} by \dfrac{8}{3}
| \displaystyle \dfrac{4}{5} \div \dfrac{8}{3} | \displaystyle = | \displaystyle \dfrac{4}{5} \cdot \dfrac{3}{8} | Take the reciprocal of the divisor \dfrac{8}{3} and multiply |
| \displaystyle \dfrac{4 \cdot 3}{5 \cdot 8} | \displaystyle = | \displaystyle \dfrac{12}{40} | Multiply the numerators and denominators |
| \displaystyle = | \displaystyle \dfrac{3}{10} | Simplify the fraction |
Mixed numbers: change it into an improper fraction and proceed to use the rules for multiplying or dividing fractions.
Divide 2\dfrac{1}{2} by 1\dfrac{3}{4}
| \displaystyle 2\dfrac{1}{2} \div 1\dfrac{3}{4} | \displaystyle = | \displaystyle \dfrac{5}{2} \div \dfrac{7}{4} | Convert mixed numbers to improper fractions |
| \displaystyle \dfrac{5}{2} \div \dfrac{7}{4} | \displaystyle = | \displaystyle \dfrac{5}{2} \cdot \dfrac{4}{7} | Take the reciprocal of the divisor \dfrac{7}{4} and multiply |
| \displaystyle \dfrac{5 \cdot 4}{2 \cdot 7} | \displaystyle = | \displaystyle \dfrac{20}{14} | Multiply the numerators and denominators |
| \displaystyle = | \displaystyle \dfrac{10}{7} | Simplify the fraction |
When we have negative rational numbers, the same rules of negative and positive integers apply:
- Multiplying or dividing two positive rational numbers gives a positive result
Multiply \dfrac{2}{3} by \dfrac{3}{4}
| \displaystyle \dfrac{2}{3} \cdot \dfrac{3}{4} | \displaystyle = | \displaystyle \dfrac{2 \cdot 3}{3 \cdot 4} | Multiply the numerators and denominators |
| \displaystyle = | \displaystyle \dfrac{6}{12} | Simplify the fraction | |
| \displaystyle = | \displaystyle \dfrac{1}{2} |
- Multiplying or dividing two negative rational numbers gives a positive result
Multiply -2.5 by -4
| \displaystyle -2.5 \cdot -4 | \displaystyle = | \displaystyle -25 \cdot -4 | Multiply the numbers as if there were no decimal places |
| \displaystyle = | \displaystyle 100 | The product will have 1+0=1 decimal place | |
| \displaystyle = | \displaystyle 10.0 | Rewrite 100 with one decimal place added in | |
| \displaystyle = | \displaystyle 10 |
- Multiplying or dividing one positive and one negative rational number gives a negative result
Divide 3 \dfrac{1}{2} by -2 \dfrac{1}{4}
| \displaystyle 3 \dfrac{1}{2} \div -2 \dfrac{1}{4} | \displaystyle = | \displaystyle \dfrac{7}{2} \div \dfrac{-9}{4} | Convert to improper fractions |
| \displaystyle = | \displaystyle \dfrac{7}{2} \cdot \dfrac{4}{-9} | Take the reciprocal of the divisor | |
| \displaystyle = | \displaystyle \dfrac{7 \cdot 4}{2 \cdot -9} | Multiply the numerators and denominators | |
| \displaystyle = | \displaystyle \dfrac{28}{-18} | Simplify the fraction | |
| \displaystyle = | \displaystyle -\dfrac{14}{9} |
Examples
Example 1
Consider the expression-10\cdot \left(-2\dfrac{1}{4}\right).
Estimate the solution.
Evaulate, giving your answer as a mixed number.
Example 2
Consider the expression - 9.11 \cdot \dfrac{5}{9}.
Evaluate, writing your answer as decimals to the thousandths.
Justify your solution.
Decimals
- Multiplying: treat the numbers as whole numbers initially. After multiplying, count the total number of decimal places in the original numbers and place the decimal in the product.
- Dividing: eliminate the decimal in the divisor by multiplying it and the dividend by 10 repeatedly until the divisor is a whole number. Then use long division.
Fractions
Multiply the numerators, multiply the denominators, and simplify.
To divide, take the reciprocal of the divisor and rewrite as multiplication.
Rewrite mixed numbers as improper fractions first.
Signs with multiplication and division
If signs are opposite the answer will be negative.
If signs are the same the answer will be positive.