To simplify the expression, we will first simplify the numerator and the denominator separately, then divide the simplified numerator by the simplified denominator.

Question

To simplify the expression, we will first simplify the numerator and the denominator separately, then divide the simplified numerator by the simplified denominator.

Asked on April 24, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-24T07:18:11.794Z

To simplify the expression, we will first simplify the numerator and the denominator separately, then divide the simplified numerator by the simplified denominator. The expression is: $$ \frac{5\frac{3}{4} \div 1\frac{1}{3} - 4\frac{1}{2} \times \frac{2}{15}}{\frac{1}{2} - \frac{7}{20}} $$ Step 1: Convert all mixed numbers to improper fractions. $$ 5\frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{23}{4} $$ $$ 1\frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3} $$ $$ 4\frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2} $$ Step 2: Simplify the numerator. The numerator is $\frac{23}{4} \div \frac{4}{3} - \frac{9}{2} \times \frac{2}{15}$. First, perform the division: $$ \frac{23}{4} \div \frac{4}{3} = \frac{23}{4} \times \frac{3}{4} = \frac{23 \times 3}{4 \times 4} = \frac{69}{16} $$ Next, perform the multiplication: $$ \frac{9}{2} \times \frac{2}{15} = \frac{9 \times 2}{2 \times 15} = \frac{18}{30} $$ Simplify the fraction $\frac{18}{30}$ by dividing both numerator and denominator by their greatest common divisor, which is 6: $$ \frac{18 \div 6}{30 \div 6} = \frac{3}{5} $$ Now, perform the subtraction in the numerator: $$ \frac{69}{16} - \frac{3}{5} $$ Find a common denominator for 16 and 5, which is 80. $$ \frac{69 \times 5}{16 \times 5} - \frac{3 \times 16}{5 \times 16} = \frac{345}{80} - \frac{48}{80} = \frac{345 - 48}{80} = \frac{297}{80} $$ So, the numerator is $\frac{297}{80}$. Step 3: Simplify the denominator. The denominator is $\frac{1}{2} - \frac{7}{20}$. Find a common denominator for 2 and 20, which is 20. $$ \frac{1 \times 10}{2 \times 10} - \frac{7}{20} = \frac{10}{20} - \frac{7}{20} = \frac{10 - 7}{20} = \frac{3}{20} $$ So, the denominator is $\frac{3}{20}$. Step 4: Divide the simplified numerator by the simplified denominator. $$ \frac{\frac{297}{80}}{\frac{3}{20}} = \frac{297}{80} \div \frac{3}{20} $$ To divide by a fraction, multiply by its reciprocal: $$ \frac{297}{80} \times \frac{20}{3} $$ Simplify by canceling common factors. 20 divides into 80 four times, and 3 divides into 297 ninety-nine times. $$ \frac{297 \div 3}{80 \div 20} = \frac{99}{4} $$ Convert the improper fraction to a mixed number: $$ \frac{99}{4} = 24\frac{3}{4} $$ The simplified expression is $\boxed{\text{24\frac{3}{4}}}$. What's next?

Accepted Answer · 2026-04-24T07:18:11.794Z

To simplify the expression, we will first simplify the numerator and the denominator separately, then divide the simplified numerator by the simplified denominator. The expression is: $$ \frac{5\frac{3}{4} \div 1\frac{1}{3} - 4\frac{1}{2} \times \frac{2}{15}}{\frac{1}{2} - \frac{7}{20}} $$ Step 1: Convert all mixed numbers to improper fractions. $$ 5\frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{23}{4} $$ $$ 1\frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3} $$ $$ 4\frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2} $$ Step 2: Simplify the numerator. The numerator is $\frac{23}{4} \div \frac{4}{3} - \frac{9}{2} \times \frac{2}{15}$. First, perform the division: $$ \frac{23}{4} \div \frac{4}{3} = \frac{23}{4} \times \frac{3}{4} = \frac{23 \times 3}{4 \times 4} = \frac{69}{16} $$ Next, perform the multiplication: $$ \frac{9}{2} \times \frac{2}{15} = \frac{9 \times 2}{2 \times 15} = \frac{18}{30} $$ Simplify the fraction $\frac{18}{30}$ by dividing both numerator and denominator by their greatest common divisor, which is 6: $$ \frac{18 \div 6}{30 \div 6} = \frac{3}{5} $$ Now, perform the subtraction in the numerator: $$ \frac{69}{16} - \frac{3}{5} $$ Find a common denominator for 16 and 5, which is 80. $$ \frac{69 \times 5}{16 \times 5} - \frac{3 \times 16}{5 \times 16} = \frac{345}{80} - \frac{48}{80} = \frac{345 - 48}{80} = \frac{297}{80} $$ So, the numerator is $\frac{297}{80}$. Step 3: Simplify the denominator. The denominator is $\frac{1}{2} - \frac{7}{20}$. Find a common denominator for 2 and 20, which is 20. $$ \frac{1 \times 10}{2 \times 10} - \frac{7}{20} = \frac{10}{20} - \frac{7}{20} = \frac{10 - 7}{20} = \frac{3}{20} $$ So, the denominator is $\frac{3}{20}$. Step 4: Divide the simplified numerator by the simplified denominator. $$ \frac{\frac{297}{80}}{\frac{3}{20}} = \frac{297}{80} \div \frac{3}{20} $$ To divide by a fraction, multiply by its reciprocal: $$ \frac{297}{80} \times \frac{20}{3} $$ Simplify by canceling common factors. 20 divides into 80 four times, and 3 divides into 297 ninety-nine times. $$ \frac{297 \div 3}{80 \div 20} = \frac{99}{4} $$ Convert the improper fraction to a mixed number: $$ \frac{99}{4} = 24\frac{3}{4} $$ The simplified expression is $\boxed{\text{24\frac{3}{4}}}$. What's next?

To simplify the expression, we will first simplify the numerator and the denominator separately, then divide the simplified numerator by the simplified denominator.