Maths Simultaneuos Equations

1)
Let t be the number of $2 coins, and f be that of $5 coins.

2t + 5f = 78 ...... (1)
(t - 3) : (f + 3) = 7 : 3 ...... (2)

(2):
3(t - 3) = 7(f + 3)
3t - 9 = 7f + 21
3t - 7f = 30
6t - 14f = 60 ...... (3)

(1) x 3:
6t + 15f = 234 ...... (4)

(4) - (3):
29f = 174
f = 6

Put f = 6 into (1):
2t + 5(6) = 78
2t = 48
t = 24

Ans: She originally had 24 $2 coins and 6 $5 coints.


2)
Let a g and b g be the weight of grade A tea and that of grade B tea respectively.

a + b = 1.8 x 1000 ...... (1)
(120/100)a + (84/100)b = (a + b)(100/100) ...... (2)

(2):
120a + 84b = 100a + 100b
20a - 16b = 0
5a - 4b = 0 ...... (3)

(1):
a + b = 1800
4a + 4b = 7200 ...... (4)

(3) + (4):
9a = 7200
a = 800

Put a = 800 into (1):
(800) + b = 1800
b = 1000

Weight that the grade B tea heavier than the grade A tea
= (b - a) g
= (1000 - 800) g
= 200 g
= 0.2 kg


3)
Let p and b be the number of pencils and that of ballpoints respectively.

a : b = 3 : 2 ...... (1)
(a + 1) : (b - 1) = 5 : 3 ...... (2)

(1):
2a = 3b
6a - 9b = 0 ...... (3)

(2):
3(a + 1) = 5(b - 1)
3a + 3 = 5b - 5
3a - 5b = -8
6a - 10b = -16 ...... (4)

(3) - (4):
b = 16

Put b = 16 into (1):
a : 16 = 3 : 2
a = 24

Number of pencils and ballpoints in total in the box
= 16 + 24
= 40