Answer :
Sure! Let's go through each problem step-by-step to find the quotient in its simplest form:
4. Estimate and simplify [tex]\(\frac{3}{4} \div \frac{5}{6}\)[/tex]:
- First, find the reciprocal of [tex]\(\frac{5}{6}\)[/tex], which is [tex]\(\frac{6}{5}\)[/tex].
- Change the division into multiplication: [tex]\(\frac{3}{4} \times \frac{6}{5}\)[/tex].
- Multiply the numerators: [tex]\(3 \times 6 = 18\)[/tex].
- Multiply the denominators: [tex]\(4 \times 5 = 20\)[/tex].
- The fraction we get is [tex]\(\frac{18}{20}\)[/tex].
- Simplify [tex]\(\frac{18}{20}\)[/tex] by finding the greatest common divisor (GCD) of 18 and 20, which is 2.
- Divide both the numerator and the denominator by 2: [tex]\(\frac{18 \div 2}{20 \div 2} = \frac{9}{10}\)[/tex].
- So, the simplified quotient is [tex]\(\frac{9}{10}\)[/tex].
5. Estimate and simplify [tex]\(3 \div \frac{3}{4}\)[/tex]:
- First, find the reciprocal of [tex]\(\frac{3}{4}\)[/tex], which is [tex]\(\frac{4}{3}\)[/tex].
- Change the division into multiplication: [tex]\(3 \times \frac{4}{3}\)[/tex].
- Multiply: [tex]\(3 \times \frac{4}{3} = \frac{12}{3}\)[/tex].
- Simplify [tex]\(\frac{12}{3}\)[/tex]: Since 12 divided by 3 equals 4, the quotient is 4.
6. Estimate and simplify [tex]\(\frac{1}{2} \div \frac{3}{4}\)[/tex]:
- First, find the reciprocal of [tex]\(\frac{3}{4}\)[/tex], which is [tex]\(\frac{4}{3}\)[/tex].
- Change the division into multiplication: [tex]\(\frac{1}{2} \times \frac{4}{3}\)[/tex].
- Multiply the numerators: [tex]\(1 \times 4 = 4\)[/tex].
- Multiply the denominators: [tex]\(2 \times 3 = 6\)[/tex].
- The fraction we get is [tex]\(\frac{4}{6}\)[/tex].
- Simplify [tex]\(\frac{4}{6}\)[/tex] by finding the GCD of 4 and 6, which is 2.
- Divide both the numerator and the denominator by 2: [tex]\(\frac{4 \div 2}{6 \div 2} = \frac{2}{3}\)[/tex].
- So, the simplified quotient is [tex]\(\frac{2}{3}\)[/tex].
7. Estimate and simplify [tex]\(\frac{5}{12} \div 3\)[/tex]:
- Convert 3 into a fraction: [tex]\(\frac{3}{1}\)[/tex].
- Find the reciprocal of [tex]\(\frac{3}{1}\)[/tex], which is [tex]\(\frac{1}{3}\)[/tex].
- Change the division into multiplication: [tex]\(\frac{5}{12} \times \frac{1}{3}\)[/tex].
- Multiply the numerators: [tex]\(5 \times 1 = 5\)[/tex].
- Multiply the denominators: [tex]\(12 \times 3 = 36\)[/tex].
- The fraction we get is [tex]\(\frac{5}{36}\)[/tex].
- This fraction is already in its simplest form since 5 and 36 have no common factors other than 1.
- So, the quotient is [tex]\(\frac{5}{36}\)[/tex].
That's the complete step-by-step explanation for each division problem!
4. Estimate and simplify [tex]\(\frac{3}{4} \div \frac{5}{6}\)[/tex]:
- First, find the reciprocal of [tex]\(\frac{5}{6}\)[/tex], which is [tex]\(\frac{6}{5}\)[/tex].
- Change the division into multiplication: [tex]\(\frac{3}{4} \times \frac{6}{5}\)[/tex].
- Multiply the numerators: [tex]\(3 \times 6 = 18\)[/tex].
- Multiply the denominators: [tex]\(4 \times 5 = 20\)[/tex].
- The fraction we get is [tex]\(\frac{18}{20}\)[/tex].
- Simplify [tex]\(\frac{18}{20}\)[/tex] by finding the greatest common divisor (GCD) of 18 and 20, which is 2.
- Divide both the numerator and the denominator by 2: [tex]\(\frac{18 \div 2}{20 \div 2} = \frac{9}{10}\)[/tex].
- So, the simplified quotient is [tex]\(\frac{9}{10}\)[/tex].
5. Estimate and simplify [tex]\(3 \div \frac{3}{4}\)[/tex]:
- First, find the reciprocal of [tex]\(\frac{3}{4}\)[/tex], which is [tex]\(\frac{4}{3}\)[/tex].
- Change the division into multiplication: [tex]\(3 \times \frac{4}{3}\)[/tex].
- Multiply: [tex]\(3 \times \frac{4}{3} = \frac{12}{3}\)[/tex].
- Simplify [tex]\(\frac{12}{3}\)[/tex]: Since 12 divided by 3 equals 4, the quotient is 4.
6. Estimate and simplify [tex]\(\frac{1}{2} \div \frac{3}{4}\)[/tex]:
- First, find the reciprocal of [tex]\(\frac{3}{4}\)[/tex], which is [tex]\(\frac{4}{3}\)[/tex].
- Change the division into multiplication: [tex]\(\frac{1}{2} \times \frac{4}{3}\)[/tex].
- Multiply the numerators: [tex]\(1 \times 4 = 4\)[/tex].
- Multiply the denominators: [tex]\(2 \times 3 = 6\)[/tex].
- The fraction we get is [tex]\(\frac{4}{6}\)[/tex].
- Simplify [tex]\(\frac{4}{6}\)[/tex] by finding the GCD of 4 and 6, which is 2.
- Divide both the numerator and the denominator by 2: [tex]\(\frac{4 \div 2}{6 \div 2} = \frac{2}{3}\)[/tex].
- So, the simplified quotient is [tex]\(\frac{2}{3}\)[/tex].
7. Estimate and simplify [tex]\(\frac{5}{12} \div 3\)[/tex]:
- Convert 3 into a fraction: [tex]\(\frac{3}{1}\)[/tex].
- Find the reciprocal of [tex]\(\frac{3}{1}\)[/tex], which is [tex]\(\frac{1}{3}\)[/tex].
- Change the division into multiplication: [tex]\(\frac{5}{12} \times \frac{1}{3}\)[/tex].
- Multiply the numerators: [tex]\(5 \times 1 = 5\)[/tex].
- Multiply the denominators: [tex]\(12 \times 3 = 36\)[/tex].
- The fraction we get is [tex]\(\frac{5}{36}\)[/tex].
- This fraction is already in its simplest form since 5 and 36 have no common factors other than 1.
- So, the quotient is [tex]\(\frac{5}{36}\)[/tex].
That's the complete step-by-step explanation for each division problem!
