When x is added to each of 13, 19, 16 and 23, then the numbers so obtained, in this order, are in proportion. Then, if $5x : y :: y : (8x-4)$, and $y > 0$, what is the value of y?
Solving the Proportion Problem: Finding x
The problem states that when a number $x$ is added to each of the numbers 13, 19, 16, and 23, the resulting numbers are in proportion. This means:
$ \frac{13+x}{19+x} = \frac{16+x}{23+x} $To solve for $x$, we cross-multiply:
$ (13+x)(23+x) = (19+x)(16+x) $Expand both sides:
$ 13 \times 23 + 13x + 23x + x^2 = 19 \times 16 + 19x + 16x + x^2 $ $ 299 + 36x + x^2 = 304 + 35x + x^2 $Now, simplify the equation by canceling $x^2$ from both sides and rearranging terms:
$ 299 + 36x = 304 + 35x $ $ 36x - 35x = 304 - 299 $ $ x = 5 $So, the value of $x$ is 5.
Finding the Value of y
We are given a second proportion involving $x$ and $y$: $5x : y :: y : (8x-4)$. We are also told that $y > 0$.
This proportion can be written as:
$ \frac{5x}{y} = \frac{y}{8x-4} $Cross-multiplying gives:
$ y^2 = 5x(8x-4) $Now, substitute the value of $x=5$ into this equation:
$ y^2 = 5(5)(8(5)-4) $ $ y^2 = 25(40-4) $ $ y^2 = 25(36) $To find $y$, we take the square root of both sides:
$ y = \sqrt{25 \times 36} $ $ y = \sqrt{25} \times \sqrt{36} $ $ y = 5 \times 6 $ $ y = 30 $Since the condition is $y > 0$, the value $y=30$ is the correct solution.
Important Questions from Ratio and Proportion (Notes)
- If A exceeds B by 50% and B is less than C by 25%, then A: C is:
- For any two numbers p and q,
$(p + q) : (p - q) : pq = 7 : 1 : 60$
If $p^2 + q^2 = r^2$, then r is equal to - If A : B = 5 : 6 and B : C = 6 : 7, then A + B : B + C : A + C is :
- The height of a tree varies as the square root of its age. When the age of the tree is 324 years, its height is 19 feet. What will be the height of the tree (in feet) at the age of 81 years?
- Three friends A, B, and C started a business by investing ₹6,000, ₹8,000, and ₹10,000 respectively. If their total profit at the end of the year is ₹12,000, what is C's share?
