john had 3/5 as many boys as girls in his class. he gave each boy 4 cards and each girl7 cards. the total number of cards the boys had was 161 fewer than the number of cards the girls had. how many boys were there in john's class? 

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.

john had 3/5 as many boys as girls in his class. he gave each boy 4 cards and each girl 7 cards. the total number of cards the boys had was 161 fewer than the number of cards the girls had. how many boys were there in john's class?

Solve the ratio problem where John gives cards to boys and girls. Learn how to set up and solve systems of equations based on ratios, with a detailed step-by-step guide. This problem involves algebraic concepts such as proportions and equation solving, perfect for Grades 7-9.

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