Operations with Fractions Practice Problems & Solutions

To solve this problem, we'll begin by finding a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{4}$.
Step 1: Identify the denominators, which are 3 and 4. Multiply these to get a common denominator: $3 \times 4 = 12$.

Step 2: Convert each fraction to an equivalent fraction with the common denominator of 12.

• To convert $\frac{1}{3}$ to a denominator of 12, multiply both the numerator and the denominator by 4: $\frac{1 \times 4}{3 \times 4} = \frac{4}{12}$.
• To convert $\frac{1}{4}$ to a denominator of 12, multiply both the numerator and the denominator by 3: $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$.

Step 3: Add the resulting fractions: $\frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12}$.

Thus, the sum of $\frac{1}{3}$ and $\frac{1}{4}$ is $\frac{7}{12}$.

To solve the problem of dividing two fractions, we apply the method of multiplication by the reciprocal:

• Step 1: The given problem is $\frac{3}{4} \div \frac{5}{6}$.
• Step 2: Find the reciprocal of the divisor $\frac{5}{6}$, which is $\frac{6}{5}$.
• Step 3: Multiply the dividend $\frac{3}{4}$ by the reciprocal of the divisor: $\frac{3}{4} \times \frac{6}{5}$
• Step 4: Perform the multiplication: $\frac{3 \times 6}{4 \times 5} = \frac{18}{20}$
• Step 5: Simplify the resulting fraction $\frac{18}{20}$: $\frac{18 \div 2}{20 \div 2} = \frac{9}{10}$

Therefore, the correct result of the division $\frac{3}{4} \div \frac{5}{6}$ is $\frac{9}{10}$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the least common multiple (LCM) of the denominators 8 and 4.
• Step 2: Convert $\frac{1}{4}$ to a fraction with the denominator of 8.
• Step 3: Add the fractions $\frac{3}{8}$ and $\frac{2}{8}$.

Now, let's work through each step:
Step 1: The LCM of 8 and 4 is 8, so this will be the common denominator.
Step 2: Transform $\frac{1}{4}$ into a fraction with the denominator of 8. Multiply the numerator and the denominator by 2 to get $\frac{2}{8}$.
Step 3: Now, add the fractions with the same denominator: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$.

Therefore, the sum of $\frac{3}{8}$ and $\frac{1}{4}$ is $\frac{5}{8}$, which matches choice 4.

To solve this problem, we'll follow these steps:

• Step 1: Identify the denominators of the fractions.
• Step 2: Because the denominators are the same, add the numerators.
• Step 3: Simplify the resulting fraction if necessary.

Now, let's work through each step:
Step 1: Both fractions, $\frac{1}{4}$ and $\frac{3}{4}$, have the same denominator, 4.
Step 2: Since the denominators are the same, we can add the numerators: $1 + 3 = 4$.
Step 3: The resulting fraction is $\frac{4}{4}$, which simplifies to $1$.

Therefore, the solution to the problem is $1$.

To solve $\frac{5}{9} \div \frac{7}{18}$, we will proceed with the following steps:

• Step 1: Identify the dividend and divisor: $\frac{5}{9}$ and $\frac{7}{18}$.
• Step 2: Find the reciprocal of the divisor. The reciprocal of $\frac{7}{18}$ is $\frac{18}{7}$.
• Step 3: Multiply the dividend by the reciprocal of the divisor: $\frac{5}{9} \times \frac{18}{7}.$
• Step 4: Multiply the numerators: $5 \times 18 = 90$.
• Step 5: Multiply the denominators: $9 \times 7 = 63$.
• Step 6: The resulting fraction from the multiplication is $\frac{90}{63}.$
• Step 7: Simplify $\frac{90}{63}$. The greatest common divisor (GCD) of 90 and 63 is 9. Divide both the numerator and the denominator by 9: $\frac{90 \div 9}{63 \div 9} = \frac{10}{7}.$
• Step 8: Convert $\frac{10}{7}$ into a mixed number. Since $10$ divided by $7$ is $1$ with a remainder of $3$, this is $1\frac{3}{7}.$

So, the solution to the problem is $1\frac{3}{7}$.