Question

Three numbers are in the ratio \[3:4:5\]. If the sum of the largest and the smallest equals the sum of the third and \[52\]. Find the numbers.

Answer 1

Hint: Three numbers are in the ratio \[3:4:5\]. We will assume the common ratio between the numbers to be \[x\]. So, the three numbers will be \[3x\], \[4x\] and \[5x\] respectively. Since it is given that the sum of the largest and the smallest equals the sum of the third and \[52\]. Using this we will equate \[3x + 5x\] to \[4x + 52\] to find the numbers.

Complete step-by-step answer:
We have to find the numbers which are in the ratio \[3:4:5\]. Given that the sum of the largest and the smallest equals the sum of the third and \[52\].
As we know that a ratio indicates how many times one number contains another.
Let the common ratio between the numbers be \[x\].
Then the first number will be \[3x\], second number will be \[4x\] and third number will be \[5x\] i.e.,
\[{\text{First number}} = 3x\]
\[{\text{Second number}} = 4x\]
\[{\text{Third number}} = 5x\]
Since it is given that the sum of the largest and the smallest equals the sum of the third and \[52\] and we can see that largest is \[5x\] and smallest is \[3x\]. Therefore, we can write:
\[ \Rightarrow 3x + 5x = 4x + 52\]
On simplifying the left hand side of the above equation we get,
\[ \Rightarrow 8x = 4x + 52\]
Subtracting \[4x\] from both the sides we get
\[ \Rightarrow 8x - 4x = 52\]
On simplification,
\[ \Rightarrow 4x = 52\]
Dividing both the sides by \[4\], we get
\[ \Rightarrow x = \dfrac{{52}}{4}\]
On simplification, we get
\[ \Rightarrow x = 13\]
So, \[13\] is the common ratio between the numbers.
Therefore,
\[{\text{First number}} = 3x\]
Putting the value of \[x\], we get
\[ \Rightarrow {\text{First number}} = 3 \times 13\]
\[ = 39\]
\[{\text{Second number}} = 4x\]
\[ \Rightarrow {\text{Second number}} = 4 \times 13\]
\[ = 52\]
\[{\text{Third number}} = 5x\]
\[ \Rightarrow {\text{Third number}} = 5 \times 13\]
\[ = 65\]
So, the correct answer is “\[39\], \[52\] and \[65\]”.

Note: A ratio indicates how many times one number contains another. The numbers in the ratio may be quantities of any kind. In most contexts, both numbers are restricted to be positive. A ratio written as \['a{\text{ }}to{\text{ }}b'\] or \['a:b'\] may be specified either by giving both constituting numbers or by just giving the value of their quotient \[\dfrac{a}{b}\].