A group for social work has 64 boys and 40 girls. During a membership drive, the same number of boys and girls join the group. If after this the ratio of boys to girls becomes 4 : 3, how many members does the group have now in total?

Introduction / Context:
This question deals with adjusting a ratio when equal numbers are added to each category. The group has boys and girls, and after adding the same number of each, the overall ratio changes to a new given value. We must use algebra to determine how many new members joined and then compute the new total membership.

Given Data / Assumptions:
• Initial number of boys = 64.
• Initial number of girls = 40.
• The same number of boys and girls join the group; let this number be x.
• After this, the ratio of boys to girls becomes 4 : 3.
• We need the new total number of members in the group.

Concept / Approach:
When equal numbers are added to both categories in a ratio, we introduce a variable x representing the additional boys and the same x additional girls. We then express the new ratio (64 + x) : (40 + x) and set it equal to 4 : 3. Solving this proportion equation for x gives the number of new boys and girls added, from which we can easily compute the total membership.

Step-by-Step Solution:
Step 1: Let x boys and x girls join. New boys = 64 + x, new girls = 40 + x.Step 2: The final ratio is (64 + x) : (40 + x) = 4 : 3.Step 3: Set up proportion: (64 + x)/(40 + x) = 4/3.Step 4: Cross-multiply: 3(64 + x) = 4(40 + x).Step 5: Expand: 192 + 3x = 160 + 4x.Step 6: Rearrange: 192 − 160 = 4x − 3x ⇒ 32 = x.Step 7: New number of boys = 64 + 32 = 96; new number of girls = 40 + 32 = 72.Step 8: New total members = 96 + 72 = 168.

Verification / Alternative check:
Check the final ratio: 96 : 72. Divide both terms by 24: 96 / 24 = 4, 72 / 24 = 3. The simplified ratio is indeed 4 : 3, matching the condition given. This confirms that adding 32 boys and 32 girls is consistent, making the new total membership 168.

Why Other Options Are Wrong:
Totals 201, 147 and 154 would correspond to different values of x, and if you plug those back into (64 + x)/(40 + x), the resulting ratio is not 4 : 3. Therefore, those totals contradict the given ratio condition.

Common Pitfalls:
Some students incorrectly add 4 and 3 to the original numbers or assume a fixed number based on guesswork, rather than setting up the proper algebraic equation. Another mistake is to forget that the same number is added to both boys and girls. Expressing the new numbers clearly as 64 + x and 40 + x and using ratio equality avoids such errors.

Final Answer:
The group now has a total of 168 members.