In a mixture of three varieties of tea, the weights are in the ratio 4 : 5 : 8. If 5 kg of the first variety, 10 kg of the second variety and some quantity of the third variety are added, the ratio of the weights becomes 5 : 7 : 9. What is the quantity (in kg) of the third variety in the new mixture?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 45
The first two varieties are fully known after the additions, so their new ratio 5 : 7 gives the multiplier of the old ratio; the third variety then follows from 5 : 7 : 9.
Old weights: \(4k\), \(5k\), \(8k\) kg
New first and second: \(4k+5\) and \(5k+10\)
\(\displaystyle \frac{4k+5}{5k+10}=\frac{5}{7}\), so \(28k+35=25k+50\)
\(3k=15\), so \(k=5\)
New first \(=4\times5+5=25\) kg, which is 5 parts, so 1 part \(=5\) kg
New third \(=9\times5=45\) kg
The new mixture has 45 kg of the third variety.
Why the other options are incorrect
Option (1) 40 — Incorrect. 40 kg is the third variety in the old mixture (\(8\times5\)), before the 5 kg was added.
Option (3) 50 — Incorrect. With 50 kg the new ratio would be 25 : 35 : 50, which is 5 : 7 : 10.
Option (4) 48 — Incorrect. 48 is not 9 parts of 5 kg; the ratio would not be 5 : 7 : 9.