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A man spends 30% of his income on rent, 20% of the remainder on food, and the rest 5600 is saved. Find his total income.
10000
12000
14000
16000
10000
Use the multiplier chain: Income ├Ч (0.70) ├Ч (0.80) = 5600. Simplify to Income ├Ч 0.56 = 5600, which clearly gives 10000.
Rent expenditure is 30% of total income, food expenditure is 20% of the remainder, and the remaining savings amount to 5600.
X├Ч(1тИТ100r1тАЛтАЛ)├Ч(1тИТ100r2тАЛтАЛ)=S
Use the multiplier chain: Income ├Ч (0.70) ├Ч (0.80) = 5600. Simplify to Income ├Ч 0.56 = 5600, which clearly gives 10000.
Students often incorrectly calculate the total expenditure as 30% + 20% = 50% and subtract 50% from income, ignoring that the 20% applies to the remainder, not the original total.
Define the income variable
Let the total income be x.
x
Calculate the remainder after rent
The man spends 30% on rent, leaving 70% of x.
x├Ч0.70=0.7x
Calculate the remainder after food
He spends 20% of the remainder (0.7x) on food, meaning 80% of the remainder is left.
0.7x├Ч0.80=0.56x
Solve for x
Equating the remaining amount to the savings value of 5600: 0.56x=5600. Dividing both sides by 0.56 gives x=0.565600тАЛ.
x=10000
A is correct because the sequential deduction of 30% and 20% of the remainder results in 56% of the total income being left, which equals 5600 when the total income is 10000.
This concept of 'successive remainder' is identical to the calculation of 'successive discounts' in Profit and Loss or 'population growth/depreciation' in Compound Interest.