Given:
The numbers 104, 34, 110 and 36
Concept used:
If a, b, c and d are four numbers and they are in continued proportion
Then a ∶ b ∶∶ c ∶ d
⇒ ad = cb
⇒ a/b = c/d
Calculation
Let the number x added
Now, according to question,
(104 + x) ∶ (34 + x) ∶∶ (110 + x) ∶ (36 + x)
⇒ (104 + x) × (36 + x) = (110 + x) × (34 + x)
⇒ 3744 + 36x + 104x + x2 = 3740 + 34x + 110x + x2
⇒ 3744 + 140x = 3740 + 144x
⇒ 4x = 4
Then x = 1
∴ The required number is 1.
Alternate Method
Calculations:
Let the number x added
Now, according to question,
(104 + x) ∶ (34 + x) ∶∶ (110 + x) ∶ (36 + x)
⇒ (104 + x)/(34 + x) = (110 + x)/(36 + x)
Let’s check options one by one,
For 1 option,
After adding 9 to each number, numbers will be 113, 43, 119 and 45
For them to be in continued proportion this ratio must be equal
⇒ 113/43 ≠ 119/45
Hence option 1 is not correct
For 2 option,
After adding 3 to each number, numbers will be 107, 37, 113 and 39
For them to be in continued proportion this ratio must be equal
⇒ 107/37 ≠ 113/39
Hence option 2 is not correct
For 3 option,
After adding 1 to each number, numbers will be 105, 35, 111 and 37
For them to be in continued proportion this ratio must be equal
⇒ 105/35 = 111/37 = 3
Hence option 3 is correct
For 4 option,
After adding 4 to each number, numbers will be 108, 38, 114 and 40
For them to be in continued proportion this ratio must be equal
⇒ 108/38 ≠ 114/40
Hence option 4 is not correct
∴ The correct answer is option 3.
Define x:
Let x be the number that we need to add to each number
Solve x:
Sequence = 10, 18, 22, 38
(term 1 + x) /(term 2 + x) = (term 3 + x) / term 4 + x)
(10 + x) / (18 + x) = (22 + x) / (38 + x)
(10 + x)(38 + x) = (18 + x) (22 + x)
380 + 10x + 38x + x² = 396 + 18x + 22x + x²
380 + 48x + x² = 396 + 40x + x²
48x - 40x = 396 - 380
8x = 16
x = 2
Answer: 2 must be added the numbers