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1. 3/ of the pupils in a lecture hall are boys. There are 374 more girls than boys. How many pupils are there in the lecture hall?

2. 2/9 of the participants in a cross-country race are boys. 3/7 of the remaining runners are men. 1/4 of the rest are girls. There are 42 more women than girls. How many participants are there in the cross-country race?

Sagot :

1) Total pupils: x

    Boys: 3/8x

    Girls:  3/8x + 374

Add:
3/8x + (3/8x + 374) = x
3/8x + 3/8x + 374 = x
6/8x + 374 = x
x - 6/8x = 374
x - 3/4 x = 374
4/4 x - 3/4 x = 374
1/4 x = 374

(4/1)(1/4) x = 374 (4/1)
x = 1496

There are 1,496 pupils in the classroom.

Boys: 3/8 (1,496) = 561 boys
Girls: 561 + 374 = 935 girls

Check:
561 + 935 = 1, 496 pupils

2) Total: x
    Boys: 2/9x

    Men: 3/7 (9/9x - 2/9x) = 3/7 (7/9x) = 3/9x

    Girls:  1/4 [9/9x - (2/9x + 3/9x)]
               = 1/4 [9/9x - 5/9x]
               = 1/4 (4/9x) 
               = 1/9x

     Women: 1/9x + 42

Add:
2/9x + 3/9x + 1/9x + 1/9x + 42 = x
7/9x + 42 = x
x - 7/9x = 42
9/9x - 7/9x = 42
2/9x = 42

(9/2) 2/9x = 42 (9/2)
x = 189

There are 189 participants in the cross-country race.

Boys: 2/9 (x) = 2/9 (189) = 42  boys
Men:  3/9 (x) = 3/9 (189) = 63  men
Girls:  1/9 (x) = 1/9 (189) = 21 girls
Women: 21 + 42 = 63 girls

Check:
42 + 63 + 21 + 63 = 189
189 = 189