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Difficulty: Very hardFractions and Decimals

A district officer allocates a development fund among four sectors: Health, Education, Infrastructure, and Agriculture. Health receives a fraction equal to 0.16‾0.1\overline{6} of the total fund. Education receives 38\frac{3}{8} of the remaining fund after the Health allocation. Infrastructure is directly allocated 0.43‾0.4\overline{3} of the original total fund. The remaining amount is given to Agriculture. If the amount allocated to Infrastructure exceeds the amount allocated to Education by ₹ 87,00087,000, what is the total development fund in Rupees?

Answer: 720000 Rupees

Answer

The total development fund is ₹ 720,000.
Converting the recurring decimals gives Health's share as 16\frac{1}{6} and Infrastructure's share as 1330\frac{13}{30}. The remaining fund after Health allocation is 56\frac{5}{6}, making Education's share equal to 38×56=516\frac{3}{8} \times \frac{5}{6} = \frac{5}{16} of the total. The difference between Infrastructure's and Education's shares is 1330−516=29240\frac{13}{30} - \frac{5}{16} = \frac{29}{240}. Equating 29240\frac{29}{240} of the total fund to ₹ 87,000 yields the total fund as ₹ 720,000.

Step-by-Step Solution

1
Convert the recurring decimals into simplified vulgar fractions.
0.16‾=16−190=1590=160.1\overline{6} = \frac{16-1}{90} = \frac{15}{90} = \frac{1}{6} and 0.43‾=43−490=3990=13300.4\overline{3} = \frac{43-4}{90} = \frac{39}{90} = \frac{13}{30}.
Converting mixed recurring decimals to fractions allows exact algebraic calculations.
2
Calculate the fraction of the total fund received by Education.
Remaining fund after Health = 1−16=561 - \frac{1}{6} = \frac{5}{6}. Education's fraction = 38×56=516\frac{3}{8} \times \frac{5}{6} = \frac{5}{16}.
Education receives a fraction of the remaining fund, not the original total fund.
3
Find the difference between Infrastructure's fraction and Education's fraction.
1330−516=104−75240=29240\frac{13}{30} - \frac{5}{16} = \frac{104 - 75}{240} = \frac{29}{240}.
The LCM of denominators 30 and 16 is 240.
4
Set up the linear equation with the given monetary difference to find the total fund FF.
29240×F=87,000  ⟹  F=87,000×24029=3,000×240=720,000\frac{29}{240} \times F = 87,000 \implies F = \frac{87,000 \times 240}{29} = 3,000 \times 240 = 720,000.
Solving for FF yields the original total fund allocation.

Key Concept

Converting mixed recurring decimals to vulgar fractions and evaluating multi-step compound fractional parts of a total quantity.
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Topics

Fractions and Decimals

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