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Work Shown:
T = total amount in savings
They split the savings in the ratio of 3:4
Fred has 3 parts, George has 4 parts
There are 3+4 = 7 parts total
Fred has 3/7 of the share and George has 4/7 of the share
Fred has 3T/7 pounds. George has 4T/7 pounds
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x = cost of present
They split the cost in the ratio 2:3, which means Fred pays 2/5 of the price and George pays 3/5 of the price
Fred pays 2x/5 pounds
George pays 3x/5 pounds
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After buying the present,
Fred has (3T/7)-(2x/5) pounds left over. Fred spent half of his money, so
(final amount) = (1/2)*(original amount)
(3T/7)-(2x/5) = (1/2)*(3T/7)
-(2x/5) = 3T/14-3T/7
-(2x/5) = 3T/14-6T/14
-(2x/5) = -3T/14
2x/5 = 3T/14
3T/14 = 2x/5
T = (2x/5)*(14/3)
T = 28x/15 ... we'll use this later
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After buying the present,
George has (4T/7)-(3x/5) pounds left over. This is equal to 21 pounds as this is the given amount George has left over.
(4T/7)-(3x/5) = 21
(4/7)*(T)-(3x/5) = 21
(4/7)*(28x/15)-(3x/5) = 21 ... plug in T = 28x/15
112x/105 - 3x/5 = 21
16x/15-3x/5 = 21
16x/15-9x/15 = 21
7x/15 = 21
7x = 21*15
7x = 315
x = 315/7
x = 45
The present costs 45 pounds
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Extra info:
If we split the cost of the present in the ratio 2:3, then,
Fred pays (2/5)*45 = 18 pounds
George pays (3/5)*45 = 27 pounds
Since Fred paid 18 pounds and he spent half his money, then his original amount of money was 36 pounds
George pays 27 pounds and has 21 pounds left over, so 21+27 = 48 pounds is the amount of money he starts with.
36+48 = 84 pounds is the original total in the savings account
note how 36/84 = 3/7 and 48/84 = 4/7, so this shows the two people have the proper ratio of splitting up the money, and it helps confirm we have the right answer.
Optionally, we found T = 28x/15 earlier and since x = 45, we can say
T = 28*x/15 = 28*45/15 = 84
