Question 12 asks students to identify the specific errors made by Ava and Adrian when solving fraction problems.

Problem Analysis:

We are tasked with identifying and explaining the mistakes made by Ava and Adrian in their fraction calculations.

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Part (a): Ava's Mistake

Ava's answer is:
\[
\frac{3}{5} + \frac{4}{7} = \frac{7}{12}
\]

#### Step 1: Understanding the Correct Method for Adding Fractions
To add two fractions, we need a common denominator. The general steps are:
1. Find the least common denominator (LCD) of the two fractions.
2. Rewrite each fraction with the LCD as the denominator.
3. Add the numerators while keeping the denominator the same.
4. Simplify the resulting fraction if possible.

#### Step 2: Identify Ava's Mistake
Ava simply added the numerators and denominators separately:
\[
\frac{3}{5} + \frac{4}{7} \neq \frac{3+4}{5+7} = \frac{7}{12}
\]
This is incorrect because adding fractions does not involve adding the denominators. Instead, the denominators must be made the same before adding the numerators.

#### Step 3: Correct Calculation
The correct way to add these fractions is:
1. Find the LCD of 5 and 7. Since 5 and 7 are coprime, their LCD is \(5 \times 7 = 35\).
2. Rewrite each fraction with the denominator 35:
\[
\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}
\]
\[
\frac{4}{7} = \frac{4 \times 5}{7 \times 5} = \frac{20}{35}
\]
3. Add the fractions:
\[
\frac{21}{35} + \frac{20}{35} = \frac{21 + 20}{35} = \frac{41}{35}
\]
4. The result is already in simplest form.

#### Final Answer for Part (a):
Ava's mistake was adding the numerators and denominators separately, which is incorrect. The correct answer is:
\[
\boxed{\frac{41}{35}}
\]

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Part (b): Adrian's Mistake

Adrian's answer is:
\[
5 \times \frac{2}{7} = \frac{10}{35}
\]

#### Step 1: Understanding the Correct Method for Multiplying a Whole Number by a Fraction
To multiply a whole number by a fraction, we:
1. Treat the whole number as a fraction with a denominator of 1.
2. Multiply the numerators together.
3. Multiply the denominators together.
4. Simplify the resulting fraction if possible.

#### Step 2: Identify Adrian's Mistake
Adrian correctly multiplied the numerator \(5 \times 2 = 10\), but he also multiplied the denominator \(1 \times 7 = 7\) incorrectly as 35. The denominator should remain 7, not 35.

#### Step 3: Correct Calculation
The correct way to multiply is:
\[
5 \times \frac{2}{7} = \frac{5 \times 2}{1 \times 7} = \frac{10}{7}
\]
The result is already in simplest form.

#### Final Answer for Part (b):
Adrian's mistake was multiplying the denominator by 7 instead of keeping it as 7. The correct answer is:
\[
\boxed{\frac{10}{7}}
\]

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Final Answers:

- Part (a): Ava's mistake was adding the numerators and denominators separately. The correct answer is \(\boxed{\frac{41}{35}}\).
- Part (b): Adrian's mistake was multiplying the denominator by 7 instead of keeping it as 7. The correct answer is \(\boxed{\frac{10}{7}}\).