About a Primary 5 Problem

I came across this P5 question:

There are altogether 405 boys and girls. Each boy is given 35 sweets, each girl 23 sweets. In total, boys have 255 more sweets than girls. How many boys are there?

I can solve using algebra and trial-and-error, but the students (my nephew) was taught to apply this 'formula':

If it were all girls: 405 X 23 = 9315 sweets
Add the 255 difference: 9315 + 255 = 9570
Each girl and boy has 23 + 35 = 58 sweets
No. of boys = 9570 / 58 = 165

I can see that mathematically it works, but what kind of method is this, and how should this be taught to the students (by applying the formula blindly?) This was never taught to us during our school days???

Chuck

I hope this is not taught as a formula! This formula is not general enough to warrant students learning it. there are so many problems and it is not productive to teach kids formula for each problem type.

I would not really if they were all girls. It is more like suppose everyone boys and girls each was given 23 sweets first.

There are altogether 405 boys and girls. Each boy is given 35 sweets, each girl 23 sweets. In total, boys have 255 more sweets than girls. How many boys are there?

If you use an algebraic method, you might let b = number of boys and set up this equation: 35b - 23(405 - b) = 255 which simplifies to 35b - 23 x 405 + 25b = 255 or 58b = 23 x 405 + 255 and hence b = [23 x 405 + 255] / 58. This is exactly the method that the teacher taught the class.

Question: How do we explain the algebraic solution? Why 23 x 405? Why plus 255? Finally why divide by 58 to get the number of boys?

Dividing a number by 58 to get the number of boys suggest that each boy got 58 sweets. But each received only 35 according to the problem! Why?

This method is actually one that many high-achieving students use and it is actually quite interesting and, at least to me, impressive that an 11-year old or 12-year old is capable of.

What is done is to give each child, boys and girls, 23 sweets. Thus 23 x 405 sweets are given out. Each girl got 23. Each boy also got 23. The 255 comes from the number of sweets the boys got (when each got 35) minus the number of sweets the girls got (when each got 23). So 23 x 405 + 255 tells us 23 x 405 (the number given to every child when each got 23) + 35b (the number the boys got when each got 35) - the number the girls got (when each got 23). The result gives number of sweets that the boys got (23, at first) + 35.

See?

This method is quite clever in that if pretends to give each boy 23 first before the 35 that the problem requires. It also explains the algebraic solution that most adults are familiar with.