Question

What number must be added to $3,5,7,10$ each in order to get four numbers in proportion?

Answer 1

Hint: First, we need to know about the concept of the proportion, which is that the product of the extreme is equal to the product of the mean. The numbers are represented as $a:b::c:d$ where $a,d$ are the extremes and $b,c$ are the mean. Using the information, we will solve the given problem.

Complete step-by-step solution:
Since from the given that we have to find the number which must be added to $3,5,7,10$ each in order to get four numbers in proportion.
Let $x$ be the unknown number that needs to be added to get the proportion.
Hence, the numbers are $3 + x,5 + x,7 + x,10 + x$ are the four numbers which is a form of proportion.
Thus, this can be represented as $a:b::c:d = 3 + x:5 + x::7 + x:10 + x$
Now the means and extremes property of the proportionality where the product of the extremes equals the product of the mean.
Hence, we have,
$(3 + x)(10 + x) = (5 + x)(7 + x)$
Further solving we have,
$(3 + x)(10 + x) = (5 + x)(7 + x) $
$\Rightarrow {x^2} + 13x + 30 = {x^2} + 12x + 35$
Canceling the common terms, we have,
 $13x - 12x = 35 - 30$
$ \Rightarrow x = 5$
Hence, we get the unknown number as $5$.

Note: The number is proportional when the ratio of the LHS of the proportional is equal to the RHS of the proportional.
We also used the basic operations, which are addition and multiplication.
Since multiplicand refers to the number multiplied. Also, a multiplier refers to the number that multiplies the first number. Have a look at an example; while multiplying $5 \times 7$ the number $5$ is called the multiplicand and the number $7$ is called the multiplier, Addition is the summing of two or more than two numbers in addition, if we sum the two or more numbers a new frame of the number will be found.