How to Multiply and Divide Rational Numbers
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Rational numbers, the linchpin of mathematical classification, symbolize values that can be articulated as a clear ratio of two integers. The vital reminder here is that the denominator, the divisor in the ratio, should never be zero, as that would yield an undefined value, a disruption to the standardized rules of mathematics. In this discussion, we will unravel the process of multiplying and dividing rational numbers.
Navigating the Process of Multiplying and Dividing Rational Numbers
Let's embark on the step-by-step exploration of multiplying and dividing rational numbers:
Phase 1: Rational Numbers - A Recap
Starting our numerical adventure, the first milestone is understanding rational numbers. These are values expressible as a quotient of two integers, with the denominator not being zero. Rational numbers include integers, fractions, and decimals that either terminate or repeat.
Phase 2: Transition to Fraction Form
To multiply or divide effectively, all rational numbers need to be in fraction form. This step includes transforming whole numbers and decimals into fractions. A whole number can be presented as a fraction over 1, while decimals can be converted with appropriate denominators like 10,100,1000, etc., based on decimal places.
Phase 3: Rational Numbers Multiplication
The multiplication process for rational numbers is quite direct: multiply the numerators to obtain the new numerator and do the same with the denominators for the new denominator. It's key to note that the product of two rational numbers remains a rational number.
Let’s take the example of two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\)
Now, \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{ac}{bd} \)
Phase 4: Rational Numbers Division
Division, a bit more nuanced, entails an extra step. When dividing by a rational number, we multiply by its reciprocal, which is attained by swapping the numerator and denominator. Hence, dividing rational numbers essentially becomes a multiplication operation.
If we have two fractions, say \(\frac{a}{b}\) and \(\frac{c}{d}\), and we want to divide the first by the second, we swap the numerator and the denominator of the second fraction to get its reciprocal, which is \(\frac{d}{c}\). So, the division of \(\frac{a}{b}\) by \(\frac{c}{d}\) is the same as the multiplication of \(\frac{a}{b}\) by the reciprocal of \(\frac{c}{d}\), which is \(\frac{d}{c}\). Mathematically, this is represented as: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}= \frac{ad}{bc} \)
Phase 5: Preliminary Simplification
To make multiplication and division easier, especially with larger numbers, it can be helpful to simplify beforehand. This simplification process involves cross-canceling any common factors in the numerator of one fraction with the denominator of the other.
Phase 6: Performing the Operation and Result Simplification
Once the fractions are prepared, you can execute the multiplication or division. After obtaining the result, strive to simplify the fraction to its simplest form if possible. This step involves dividing the numerator and the denominator by any common factors until none remain.
Phase 7: Managing Negative Signs
Post-operation, it's essential to handle the signs properly. A rational number is negative if either (but not both) the numerator or the denominator is negative. Consequently, the negative sign can be positioned with the numerator, the denominator, or in front of the fraction.
Phase 8: Conversion (If Needed)
The final step, based on context or problem requirements, may involve converting the fraction back to a decimal form or a mixed number.
Exercises for Multiplying and Dividing Rational Numbers
1) \({2 \over 3} \ \times \ {17 \over 15} = \)
2) \({3 \over 8} \ \times \ {10 \over 16} = \)
3) \({4 \over 6} \ \times \ {18 \over 17} = \)
4) \({7 \over 10} \ \times \ {7 \over 8} = \)
5) \({7 \over 3} \ \times \ {1 \over 5} = \)
6) \({9 \over 7} \ \times \ {6 \over 5} = \)
7) \({10 \over 5} \ \div \ {8 \over 3} = \)
8) \({6 \over 3} \ \div \ {14 \over 18} = \)
9) \({6\over 10} \ \div \ {16 \over 17} = \)
10) \({8\over 7} \ \div \ {15 \over 20} = \)
11) \({2\over 5} \ \div \ {3 \over 2} = \)
12) \({1\over 6} \ \div \ {3 \over 6} = \)
1)\({2 \over 3} \ \times \ {17 \over 15} = \)\( \ \color{red}{{2 \times 17 \over 3\times15} = } \) \(\color{red}{{34 \over 45}}\)
Solution:
Multiply the top numbers, and then multiply the bottom numbers.
\({2 \over 3} \ \times \ {17 \over 15} = \)\({2 \times 17 \over 3\times15} = \) \({34 \over 45}\)
2)\({3 \over 8} \ \times \ {10 \over 16} = \)\( \ \color{red}{{3 \times 10 \over 8\times16} = } \) \(\color{red}{{30 \over 128}}\)\(\color{red}{ = {30 \div 2 \over 128 \div 2} = {15 \over 64}} \)
GCF(30,128) = 2
Solution:
Step1: Multiply the top numbers, and then multiply the bottom numbers.
\({3 \over 8} \ \times \ {10 \over 16} = \)\( {3 \times 10 \over 8\times16} = \) \({30 \over 128}\)
Step 2: Simplify your answer. \({30 \over 128}\)\( = {30 \div 2 \over 128 \div 2} = {15 \over 64} \)
3)\({4 \over 6} \ \times \ {18 \over 17} = \)\( \ \color{red}{{4 \times 18 \over 6\times17} = } \) \(\color{red}{{72 \over 102}}\)\(\color{red}{ = {72 \div 6 \over 102 \div 6} = {12 \over 17}} \)
GCF(72,102) = 6
Solution:
Step1: Multiply the top numbers, and then multiply the bottom numbers.
\({3 \over 8} \ \times \ {10 \over 16} = \)\({3 \times 10 \over 8\times16} = \) \({30 \over 128}\)
Step 2: Simplify your answer. \({72 \over 102}\)\( = {72 \div 6 \over 102 \div 6} = {12 \over 17}\)
4)\({7 \over 10} \ \times \ {7 \over 8} = \)\( \ \color{red}{{7 \times 7 \over 10\times8} = } \) \(\color{red}{{49 \over 80}}\)
Solution:
Step1: Multiply the top numbers, and then multiply the bottom numbers.
\({7 \over 10} \ \times \ {7 \over 8} = \)\({7 \times 7 \over 10\times8} = \) \({49 \over 80}\)
5)\({7 \over 3} \ \times \ {1 \over 5} = \)\( \ \color{red}{{7 \times 1 \over 3\times5} = } \) \(\color{red}{{7 \over 5}}\)
Solution:
Step1: Multiply the top numbers, and then multiply the bottom numbers.
\({7 \over 3} \ \times \ {1 \over 5} = \)\({7 \times 1 \over 3\times5} =\) \({7 \over 5}\)
6)\({9 \over 7} \ \times \ {6 \over 5} = \)\( \ \color{red}{{9 \times 6 \over 7\times5} = } \) \(\color{red}{{54 \over 35}}\)
Solution:
Step1: Multiply the top numbers, and then multiply the bottom numbers.
\({9 \over 7} \ \times \ {6 \over 5} = \)\({9 \times 6 \over 7\times5} = \) \({54 \over 35}\)
7)\({10 \over 5} \ \div \ {8 \over 3} = \)\({10 \over 5} \ \times \ {3 \over 8} = \)\( \ \color{red}{{10 \times 3 \over 5\times8} = } \) \(\color{red}{{30 \over 40}}\)\(\color{red}{ = {30 \div 10 \over 40 \div 10} = {3 \over 4}} \)
GCF(30,40) = 10
Solution:
Step1: Keep first fraction, change division sign to multiplication, and flip the numerator and denominator of the second fraction. Then multiply them.
\({10 \over 5} \ \div \ {8 \over 3} = \)\({10 \over 5} \ \times \ {3 \over 8} = \)\({10 \times 3 \over 5\times8} =\) \({30 \over 40}\)
Step 2: Simplify your answer. \({30 \over 40}\)\( = {30 \div 10 \over 40 \div 10} = {3 \over 4} \)
8)\({6 \over 3} \ \div \ {14 \over 18} = \)\({6 \over 3} \ \times \ {18 \over 14} = \)\( \ \color{red}{{6 \times 18 \over 3\times14} = } \) \(\color{red}{{108 \over 42}}\)\(\color{red}{ = {108 \div 6 \over 42 \div 6} = {18 \over 7}}\)
GCF(108,42) = 6
Solution:
Step1: Keep first fraction, change division sign to multiplication, and flip the numerator and denominator of the second fraction. Then Multiply them:
\({6 \over 3} \ \div \ {14 \over 18} = \)\({6 \over 3} \ \times \ {18 \over 14} = \)\({6 \times 18 \over 3\times14} = \) \({108 \over 42}\)
Step 2: Simplify your answer. \({108 \over 42}\)\( = {108 \div 6 \over 42 \div 6} = {18 \over 7} \)
9)\({6\over 10} \ \div \ {16 \over 17} = \)\({6 \over 10} \ \times \ {17 \over 16} = \)\( \ \color{red}{{6 \times 17 \over 10\times 16} = } \)\(\color{red}{{102 \over 160}}\)\(\color{red}{ = {102 \div 2\over 160\div 2} = {51 \over 80}}\)
GCF(102,160) = 2
Solution:
Step1: Keep first fraction, change division sign to multiplication, and flip the numerator and denominator of the second fraction. Then Multiply them:
\({6\over 10} \ \div \ {16 \over 17} = \)\({6 \over 10} \ \times \ {17 \over 16} = \)\({6 \times 17 \over 10\times 16} = \)\({102 \over 160}\)
Step 2: Simplify your answer. \({102 \over 160}\)\( = {102 \div 2\over 160\div 2} = {51 \over 80}\)
10)\({8\over 7} \ \div \ {15 \over 20} = \)\({8 \over 7} \ \times \ {20 \over 15} = \)\( \ \color{red}{{8 \times 20 \over 7\times 15} = } \)\(\color{red}{{160 \over 105}}\)\(\color{red}{ = {160 \div 5\over 105\div 5} = {32 \over 21}} \)
GCF(160,105) = 5
Solution:
Step1: Keep first fraction, change division sign to multiplication, and flip the numerator and denominator of the second fraction. Then Multiply them:
\({8\over 7} \ \div \ {15 \over 20} = \)\({8 \over 7} \ \times \ {20 \over 15} = \)\({8 \times 20 \over 7\times 15= } \)\({160 \over 105}\)
Step 2: Simplify your answer. \({160 \over 105}\)\( = {160 \div 5\over 105\div 5} = {32 \over 21}\)
11)\({2\over 5} \ \div \ {3 \over 2} = \)\({2 \over 5} \ \times \ {2 \over 3} = \)\( \ \color{red}{{2 \times 2 \over 5\times3} = } \)\(\color{red}{{4 \over 15}}\)
Solution:
Step1: Keep first fraction, change division sign to multiplication, and flip the numerator and denominator of the second fraction. Then Multiply them:
\({2\over 5} \ \div \ {3 \over 2} = \)\({2 \over 5} \ \times \ {2 \over 3} = \)\({2 \times 2 \over 5\times3} = \)\({4 \over 15}\)
12)\({1\over 6} \ \div \ {3 \over 6} = \)\({1 \over 6} \ \times \ {6 \over 3} = \)\( \ \color{red}{{1 \times 6 \over 6\times 3} = } \)\(\color{red}{{6 \over 18}}\)\(\color{red}{ = {6 \div 6\over 18\div 6} = {1 \over 3}} \)
GCF(6,18) = 6
Solution:
Step1: Keep first fraction, change division sign to multiplication, and flip the numerator and denominator of the second fraction. Then Multiply them:
\({1\over 6} \ \div \ {3 \over 6} = \)\({1 \over 6} \ \times \ {6 \over 3} = \)\({1 \times 6 \over 6\times 3} = \)\({6 \over 18}\)
Step 2: Simplify your answer. \({6 \over 18}\)\( = {6 \div 6\over 18\div 6} = {1 \over 3}\)
Multiply and Divide Rational Numbers Quiz
