At a sports carnival,the ratio of number of pupils sent by school A,B and school C to the event was 1:2:3. The number of boys from school A and the number of girls from school B were the same. The ratio of the number of girls from school C to the total number of girls was 7:10. There were 124 fewer boys than girls altogether.

(a) What was the ratio of the number of boys from school C to the total number of boys?
(b) What was the total number of girls at the carnival? ( express your answer in the simplest form)

students from A -- x

students from B -- 2x
students from C -- 3x
total students = 6x

girls from C --- 7y
total girls ------ 10y

total boys ------ 6x-10y
"There were 124 fewer boys than girls altogether"
---> girls - boys = 124
---> 10y - (6x-10y) = 124
20y - 6x = 124
10y - 3x = 62
10y = 62+3x , clearly 62+3x must be a multiple of 10
possible cases: x must end in 6
x=6 , y = 8
students from A = 6
students from B = 12
students from C = 18
total students = 36
total girls = 80
total boys = 36-80 = -44 ---- Fail

x = 16, y = 11
students from A = 16
students from B = 32
students from C = 48
total students = 96
total girls = 110
total boys = 96-110 = -14 ---- Fail

x = 26 , y = 14
students from A = 26
students from B = 52
students from C = 78
total students = 156
total girls = 140
total boys = 156-140 = 16 ---- Fail, not a difference of 124

x = 36 , y = 17
students from A = 36
students from B = 72
students from C = 108
total students = 216
total girls = 170
total boys = 216-170 = 46 ---- Fail

x = 46 , y = 20
students from A = 46
students from B = 92
students from C = 138
total students = 276
total girls = 200
total boys = 276-200 = 76 ---- Fail

x = 56 , y = 23
students from A = 56
students from B = 112
students from C = 168
total students = 336
total girls = 230
total boys = 336-230 = 106 ---- Fail

etc

cute little program, to show the extension of this

10 for x = 6 to 106 step 10
20 y = (62+3*x)/10
30 if 6*x-10*y = 124 then end
40 print x,y,6*x-10*y
50 next x
>run
6 8 -44
16 11 -14
26 14 16
36 17 46
46 20 76
56 23 106
66 26 136
76 29 166
86 32 196
96 35 226
106 38 256
>

looks like there are no values that satisfy your conditions

To solve this problem, we can start by assigning variables to the quantities mentioned in the question. Let's call the number of pupils sent by school A, B, and C as A, B, and C, respectively.

Given that the ratio of the number of pupils sent by school A, B, and C is 1:2:3, we can set up the equation:
A : B : C = 1 : 2 : 3

Next, we are told that the number of boys from school A and the number of girls from school B are the same. Let's denote the number of boys from A as BA, and the number of girls from B as GB. Therefore, the ratio of BA to GB is 1:1.

We are also given that the ratio of the number of girls from school C to the total number of girls is 7:10. Let's denote the number of girls from C as GC. Therefore, the ratio of GC to (GB + GC) is 7:10.

Now, we are told that the number of boys is 124 fewer than the number of girls altogether. This can be expressed as:
BA + BC = GB + GC - 124

To find the solution, we need to form the equations using the ratios mentioned above. Let's solve for the unknowns step by step.

First, let's assign a value to the ratio. Let's assume that A is 1, B is 2, and C is 3 (since the variables don't represent actual numbers, any values can be chosen to simplify the equations).

Since A is 1, B would be 2 * A = 2, and C would be 3 * A = 3.

Now, let's consider the ratio of boys from A to girls from B. Since the ratio is 1:1, we can assume BA is x and GB is also x.

We know that GC is 7/10 times the total number of girls (GB + GC). We can represent this as an equation:
GC = (7/10) * (GB + GC)

Next, let's consider the equation relating the number of boys and girls:
BA + BC = GB + GC - 124

Substituting the values we determined earlier:
x + BC = x + GC - 124

Simplifying the equation:
BC = GC - 124

Now, let's substitute the ratio of GC to (GB + GC) into the equation:
BC = (7/10) * (GB + BC) - 124

To solve this system of equations, we need to eliminate one of the variables. We can eliminate BC by substituting the equation BC = GC - 124 into the other equation:
(7/10) * (GB + GC) - 124 = GC - 124

Simplifying and rearranging the equation:
(7/10) * (GB + GC) = 2 * 124
7 * (GB + GC) = 20 * 124
7 * (GB + GC) = 2480
GB + GC = 2480/7

Now, let's substitute the value of GB + GC into the equation we obtained earlier:
BC = (7/10) * (GB + BC) - 124
BC = (7/10) * (2480/7) - 124
BC = 2480/10 - 124
BC = 248 - 124
BC = 124

Finally, we can find the values of BA, BC, and GC:
BA = x = BC = 124
BC = 124
GC = 2480/7 - GB

Therefore, the answers are:
(a) The ratio of the number of boys from school C to the total number of boys is 124/248 = 31/62, which can be simplified as 1/2.
(b) The total number of girls at the carnival is GB + GC = GB + (2480/7 - GB) = 2480/7.