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.
What number must be added to each of the numbers
Consider x be added to each number
So the numbers will be
Based on the condition
By cross multiplication
By further calculation
So we get
Hence,
must be added to each of the numbers.
Mathematics
Let the number to be added be x.
So, the new numbers 15 + x, 17 + x, 34 + x, 38 + x are proportional.
∴15+x17+x=34+x38+x⇒(15+x)(38+x)=(34+x)(17+x)⇒(570+15x+38x+x2)=(578+34x+17x+x2)⇒x2+53x−x2−51x=578−570⇒2x=8⇒x=4.\therefore \dfrac{15 + x}{17 + x} = \dfrac{34 + x}{38 + x} \\[1em] \Rightarrow (15 + x)(38 + x) = (34 + x)(17 + x) \\[1em] \Rightarrow (570 + 15x + 38x + x^2) = (578 + 34x + 17x + x^2) \\[1em] \Rightarrow x^2 + 53x - x^2 - 51x = 578 - 570 \\[1em] \Rightarrow 2x = 8 \\[1em] \Rightarrow x = 4.∴17+x15+x=38+x34+x⇒(15+x)(38+x)=(34+x)(17+x)⇒(570+15x+38x+x2)=(578+34x+17x+x2)⇒x2+53x−x2−51x=578−570⇒2x=8⇒x=4.
Hence, the number that must be added to each number is 4.