There were some children in a concert hall. ¹₄ of the children lined up in 8 short rows and ¹₆ of the children lined up in 4 long rows. The remaining children did not line up. There are 8 more children in each long row than each short row.
(a) What fraction of the children did not line up?
(b) How many children lined up?

	Short rows	Long rows
Number	8	4
Value	1 u	1 u + 8
Total value	8 u	4 u + 32

(a)
Fraction of children who did not line up
= 1 - ¹₄  - ¹₆  
= 1 - ³₁₂ - ²₁₂
= 1 - ⁵₁₂
= ⁷₁₂

(b)
³₁₂ of the children lined up in short rows.

8 u = 3 p --- (1)

²₁₂ of the children lined up in long rows.

4 u + 32 = 2 p --- (2)

(1)_x2
16 u = 6 p --- (3)

(2)_x3
12 u + 96 = 6 p --- (4)

(3) = (4)
16 u = 12 u + 96
16 u - 12 u = 96
4 u =  96

1 u = 96 ÷ 4 = 24

Number of children who lined up
= 8 u + 4 u + 32
= 12 u + 32
= 288 + 32
= 320

Answer(s): (a) ⁷₁₂; (b) 320