Let's define variables to solve this problem:

• Let $B$ be the number of boys in the class.
• Let $G$ be the number of girls in the class.

From the information given:

1. "John had $\frac{3}{5}$ as many boys as girls in his class." This means: $B = \frac{3}{5}G$

2. "He gave each boy 4 cards and each girl 7 cards." So, the total number of cards given to the boys is $4B$, and the total number of cards given to the girls is $7G$.

3. "The total number of cards the boys had was 161 fewer than the number of cards the girls had." This gives the equation: $4B = 7G - 161$

Step 1: Solve the system of equations

We now have two equations:

1. $B = \frac{3}{5}G$
2. $4B = 7G - 161$

Substitute $B = \frac{3}{5}G$ into the second equation: $4 \left( \frac{3}{5}G \right) = 7G - 161$ Simplify: $\frac{12}{5}G = 7G - 161$ Multiply the entire equation by 5 to eliminate the fraction: $12G = 35G - 805$ Subtract $12G$ from both sides: $0 = 23G - 805$ Add 805 to both sides: $23G = 805$ Solve for $G$: $G = \frac{805}{23} = 35$

Step 2: Solve for $B$

Using the equation $B = \frac{3}{5}G$: $B = \frac{3}{5} \times 35 = 21$

Final Answer:

There were 21 boys in John's class.

Would you like further details on any part of the solution?

Here are 5 related questions to deepen understanding:

1. How would the problem change if the ratio of boys to girls was different?
2. What if each boy had received 5 cards instead of 4? How would that affect the outcome?
3. Can you generalize a formula for similar problems involving ratios and differences in total?
4. What is the importance of setting up equations in word problems like this?
5. How can we verify the accuracy of the solution through substitution?

Tip: Always double-check your equations by substituting the variables back into the original problem to verify the results.