If (x+y):(y+z):(z+x)=7:9:13 for positive reals x,y,z, find x:y:z?
11:3:15
Introduce a common multiplier \(k\) so the given ratio becomes equations: \(x+y = 7k\), \(y+z = 9k\), \(z+x = 13k\).
Add all three. Each variable appears in exactly two of the sums, so the left side is \(2(x+y+z)\).
\(2(x+y+z) = 7k+9k+13k = 29k \Rightarrow x+y+z = 14.5k\).
Subtract each pair-sum from the total to isolate a single variable. \(x = (x+y+z)-(y+z) = 14.5k - 9k = 5.5k\).
\(y = (x+y+z)-(z+x) = 14.5k - 13k = 1.5k\), and \(z = (x+y+z)-(x+y) = 14.5k - 7k = 7.5k\).
\(x:y:z = 5.5 : 1.5 : 7.5\). Multiply each part by 2 to clear decimals: \(11 : 3 : 15\).
Key idea: summing symmetric pair-ratios gives the whole, and subtracting a pair leaves one variable. Hence \(x:y:z\) is 11:3:15.
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Important Questions from Ratio and Proportion
In a mixture of 156 litres, the ratio of milk and water is 7 : 6. How much water should be added to make the ratio 14 : 13?
If the ratio of the first to second number is 3 : 4 and that of the second to the third number is 8 : 5, and sum of three numbers is 190 then the third number is:
The third proportional to 9 and 15 is:
The average age of 3 persons is 30 years.If their ages are in the ratio of 3 : 5 : 7 respectively, then the age of the eldest person is:
The income of A and B are in the ratio 5 : 3. The expenses of A, B and C are in the ratio of 8 : 5 : 2. If C spends 2000 and B saves ₹ 700, then A saves:
