Math / Numbers & Operations

Questionb. $15 \frac{3}{8}-11 \frac{5}{12}$ c. $\frac{6}{25} \cdot \frac{5}{12}$ d. $2 \frac{1}{2} \div 3 \frac{1}{4}$ e. $\frac{2}{5}+\frac{3}{4}-\frac{1}{2}+\frac{3}{8}$

Studdy Solution

STEP 1

1. Both numbers are mixed fractions and can be converted to improper fractions.
2. Subtraction of fractions requires a common denominator.

STEP 2

1. Convert mixed fractions to improper fractions.
2. Find the least common denominator (LCD).
3. Convert fractions to have the same denominator.
4. Perform the subtraction.
5. Simplify the result if possible.
6. 1. Multiply the numerators.
2. Multiply the denominators.
3. Simplify the resulting fraction.
STEP_1: High_Level_Step: 1 Multiply the numerators.
$6 \times 5 = 30$ High_Level_Step_Completed: TRUE
STEP_2: High_Level_Step: 2 Multiply the denominators.
$25 \times 12 = 300$ High_Level_Step_Completed: TRUE
STEP_3: High_Level_Step: 3 Combine the results to form a fraction.
$\frac{30}{300}$ High_Level_Step_Completed: TRUE
STEP_4: High_Level_Step: 3 Simplify the fraction.
$\frac{30}{300} = \frac{1}{10}$ High_Level_Step_Completed: TRUE
Solution: $\frac{1}{10}$ High_Level_Step_Completed: TRUE
### Problem d: $2 \frac{1}{2} \div 3 \frac{1}{4}$
_ASSUMPTIONS_:
1. Both numbers are mixed fractions and can be converted to improper fractions.
2. Division of fractions involves multiplying by the reciprocal.
_HIGH_LEVEL_APPROACH_:
1. Convert mixed fractions to improper fractions.
2. Find the reciprocal of the divisor.
3. Multiply the dividend by the reciprocal.
4. Simplify the result if possible.
STEP_1: High_Level_Step: 1 Convert $2 \frac{1}{2}$ to an improper fraction.
$2 \frac{1}{2} = \frac{2 \cdot 2 + 1}{2} = \frac{5}{2}$ High_Level_Step_Completed: TRUE
STEP_2: High_Level_Step: 1 Convert $3 \frac{1}{4}$ to an improper fraction.
$3 \frac{1}{4} = \frac{3 \cdot 4 + 1}{4} = \frac{13}{4}$ High_Level_Step_Completed: TRUE
STEP_3: High_Level_Step: 2 Find the reciprocal of the divisor $\frac{13}{4}$ .
$\text{Reciprocal of } \frac{13}{4} \text{ is } \frac{4}{13}$ High_Level_Step_Completed: TRUE
STEP_4: High_Level_Step: 3 Multiply the dividend $\frac{5}{2}$ by the reciprocal of the divisor.
$\frac{5}{2} \times \frac{4}{13} = \frac{5 \times 4}{2 \times 13} = \frac{20}{26}$ High_Level_Step_Completed: TRUE
STEP_5: High_Level_Step: 4 Simplify the resulting fraction.
$\frac{20}{26} = \frac{10}{13}$ High_Level_Step_Completed: TRUE
Solution: $\frac{10}{13}$ High_Level_Step_Completed: TRUE
### Problem e: $\frac{2}{5} + \frac{3}{4} - \frac{1}{2} + \frac{3}{8}$
_ASSUMPTIONS_:
1. Addition and subtraction of fractions require a common denominator.
2. Simplification of the result is necessary if possible.
_HIGH_LEVEL_APPROACH_:
1. Find the least common denominator (LCD) of the fractions.
2. Convert each fraction to have the same denominator.
3. Perform the addition and subtraction.
4. Simplify the result if possible.
STEP_1: High_Level_Step: 1 Find the least common denominator (LCD) of 5, 4, 2, and 8.
The LCD of 5, 4, 2, and 8 is 40. High_Level_Step_Completed: TRUE
STEP_2: High_Level_Step: 2 Convert $\frac{2}{5}$ to have the denominator 40.
$\frac{2}{5} = \frac{2 \times 8}{5 \times 8} = \frac{16}{40}$ High_Level_Step_Completed: TRUE
STEP_3: High_Level_Step: 2 Convert $\frac{3}{4}$ to have the denominator 40.
$\frac{3}{4} = \frac{3 \times 10}{4 \times 10} = \frac{30}{40}$ High_Level_Step_Completed: TRUE
STEP_4: High_Level_Step: 2 Convert $\frac{1}{2}$ to have the denominator 40.
$\frac{1}{2} = \frac{1 \times 20}{2 \times 20} = \frac{20}{40}$ High_Level_Step_Completed: TRUE
STEP_5: High_Level_Step: 2 Convert $\frac{3}{8}$ to have the denominator 40.
$\frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40}$ High_Level_Step_Completed: TRUE
STEP_6: High_Level_Step: 3 Perform the additions and subtractions.
$\frac{16}{40} + \frac{30}{40} - \frac{20}{40} + \frac{15}{40} = \frac{16 + 30 - 20 + 15}{40} = \frac{41}{40}$ High_Level_Step_Completed: TRUE
STEP_7: High_Level_Step: 4 Simplify the result if possible.
$\frac{41}{40} \text{ is already in simplest form.}$ High_Level_Step_Completed: TRUE
Solution: $\frac{41}{40}$ High_Level_Step_Completed: TRUE

STEP 3

Convert $15 \frac{3}{8}$ to an improper fraction.
$15 \frac{3}{8} = \frac{15 \cdot 8 + 3}{8} = \frac{120 + 3}{8} = \frac{123}{8}$

STEP 4

Convert $11 \frac{5}{12}$ to an improper fraction.
$11 \frac{5}{12} = \frac{11 \cdot 12 + 5}{12} = \frac{132 + 5}{12} = \frac{137}{12}$

STEP 5

Find the least common denominator (LCD) of 8 and 12.
The LCD of 8 and 12 is 24.

STEP 6

Convert $\frac{123}{8}$ to have the denominator 24.
$\frac{123}{8} = \frac{123 \times 3}{8 \times 3} = \frac{369}{24}$

STEP 7

Convert $\frac{137}{12}$ to have the denominator 24.
$\frac{137}{12} = \frac{137 \times 2}{12 \times 2} = \frac{274}{24}$

STEP 8

Subtract the fractions.
$\frac{369}{24} - \frac{274}{24} = \frac{369 - 274}{24} = \frac{95}{24}$

STEP 9

Simplify the result if possible.
$\frac{95}{24} \text{ is already in simplest form.}$
Solution: $\frac{95}{24}$
### Problem c: $\frac{6}{25} \cdot \frac{5}{12}$
_ASSUMPTIONS_:
1. Multiplication of fractions is straightforward.
2. Simplification of the result is necessary if possible.

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