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96 is divided into two parts in such a way that a seventh part of the first and a ninth part of the second are equal. Find the larger part.
42
38
48
54
54
Use the ratio directly: parts are in ratio 7:9. The sum is 16 units. 16 units = 96, so 1 unit = 6. The larger part is 9 units = 9 ├Ч 6 = 54.
Total value = 96. First part (x) / 7 = Second part (y) / 9.
x+y=96, 7xтАЛ=9yтАЛ
Use the ratio directly: parts are in ratio 7:9. The sum is 16 units. 16 units = 96, so 1 unit = 6. The larger part is 9 units = 9 ├Ч 6 = 54.
Many students mistake the 'seventh part' to mean dividing by 7 and mistakenly equate the ratio as 9:7 instead of 7:9, leading to the smaller part result.
Establish Relationships
Let the two parts be x and y. We know x+y=96. According to the problem, the seventh part of the first equals the ninth part of the second: 7xтАЛ=9yтАЛ
x:y=7:9
Calculate Ratio Units
The ratio of the two parts is 7:9. Let the parts be 7k and 9k. Their sum is 7k+9k=16k. Since the sum is 96, we have 16k=96.
k=1696тАЛ=6
Find the Larger Part
The two parts are 7k=7├Ч6=42 and 9k=9├Ч6=54. The larger part is 54.
9├Ч6=54
D is correct because the ratio of the parts is 7:9 and the larger part equals 9 units of the total sum, resulting in 54.
This ratio method applies to all 'division of a quantity' problems in Arithmetic and Partnership chapters; always convert fractions to simple ratios to solve instantly.
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