Question

What number must be subtracted from each of the numbers $10,12,19,24$ to get the numbers which are in proportion?

Answer 1

Hint: The number subtracting from the given numbers is the same. So we can form an equation using the definition of proportion. Proportionality can be expressed using fractions. Solving this we get the answer.

Formula used: If we say four numbers $a,b,c,d$ are in proportion we mean $a:b = c:d$ or $\dfrac{a}{b} = \dfrac{c}{d}$.

Complete step-by-step solution:
We are given four numbers $10,12,19,24$.
It is said that the same number when subtracted from these numbers make a proportion.
If we say four numbers $a,b,c,d$ are in proportion we mean $a:b = c:d$ or $\dfrac{a}{b} = \dfrac{c}{d}$.
Let the number subtracting be $x$.
So we have,
$\Rightarrow$$\dfrac{{10 - x}}{{12 - x}} = \dfrac{{19 - x}}{{24 - x}}$
Cross multiplying we have,
$\Rightarrow$$(10 - x)(24 - x) = (12 - x)(19 - x)$
Opening the brackets we get,
$\Rightarrow$$240 - 10x - 24x + {x^2} = 228 - 12x - 19x + {x^2}$
Simplifying we get,
$\Rightarrow$$240 - 34x + {x^2} = 228 - 31x + {x^2}$
Cancelling ${x^2}$ from both sides we get,
$\Rightarrow$$240 - 34x = 228 - 31x$
Rearranging the terms we have,
$\Rightarrow$$240 - 228 = - 31x + 34x$
$ \Rightarrow 3x = 12$
Dividing both sides by $3$ we get,
$\Rightarrow$$x = \dfrac{{12}}{3} = 4$
That is, $4$ has to be subtracted from the numbers to make them proportion.

$\therefore $ The answer is $4$.

Note: Proportion says that two ratios (or fractions) are equal.
If $a:b = c:d$, then the quantity $d$ is called the fourth proportional to $a,b$ and $c$.
Ratio and proportion is a widely used concept in Mathematics as well as in day to day life.
If the number subtracting from each number is different we cannot form an equation like this. So we need more information for those problems. But here this works, since the same number is subtracting from each. So we got an equation with a single variable which could be solved easily.