When x is subtracted from each of 43, 38, 11 and 10, then the numbers so obtained in this order are in proportion. What is the mean proportional between (11x + 3) and (9x - 2) ?
30
Finding the Value of x from Proportion
The problem states that when a value 'x' is subtracted from each of the numbers 43, 38, 11, and 10, the resulting numbers are in proportion. This means the ratio of the first two new numbers is equal to the ratio of the last two new numbers.
We can write this relationship as:
$$ \frac{43 - x}{38 - x} = \frac{11 - x}{10 - x} $$To solve for 'x', we cross-multiply:
$$ (43 - x)(10 - x) = (38 - x)(11 - x) $$Expand both sides of the equation:
$$ 430 - 43x - 10x + x^2 = 418 - 38x - 11x + x^2 $$Combine like terms:
$$ 430 - 53x + x^2 = 418 - 49x + x^2 $$The \( x^2 \) terms cancel out on both sides. Now, we rearrange the equation to solve for 'x':
$$ 430 - 418 = 53x - 49x $$ $$ 12 = 4x $$Divide by 4:
$$ x = \frac{12}{4} $$ $$ x = 3 $$So, the value of x is 3.
Calculating Terms for Mean Proportional
Now we need to find the mean proportional between two new expressions: (11x + 3) and (9x - 2). First, let's find the values of these expressions using \( x = 3 \).
- First expression: \( 11x + 3 = 11(3) + 3 = 33 + 3 = 36 \)
- Second expression: \( 9x - 2 = 9(3) - 2 = 27 - 2 = 25 \)
Calculating the Mean Proportional
The mean proportional between two numbers, say 'a' and 'b', is defined as the square root of their product, which is \( \sqrt{ab} \).
In this case, we need to find the mean proportional between 36 and 25.
Mean Proportional = \( \sqrt{36 \times 25} \)
Calculate the product:
$$ 36 \times 25 = 900 $$Now, find the square root:
$$ \text{Mean Proportional} = \sqrt{900} $$ $$ \text{Mean Proportional} = 30 $$Therefore, the mean proportional between (11x + 3) and (9x - 2) is 30.
Important Questions from Mean Proportional
What number must be added to each of the numbers 15, 9 and 5 so that the resulting numbers may be in a continued proportion ?
Find the mean proportional between 4 and 900.
If 22, x, 88 are in a continued proportion, find the value of x.
A. 24
B. 33
C. 44
D. 36
The arithmetic mean and geometric mean of two numbers are 7 and 2√10 respectively, then find the numbers.
When x is subtracted from each of 24, 30, 36 and 46, the numbers so obtained in this order, are in proportion. What is the mean proportional between (2x + 3) and (3x + 2)?
