Question

When one is added to each of the two numbers their ratio becomes $1:2$ and when $5$ subtracted from each of them their ratio becomes $5:11$.Find the numbers.

Answer 1

Step 1:Assuming the numbers by variables $\mathit{a}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{b}$

Let the numbers be $a$and $b$ .In the ratio form, we can express it as $a:b$or $\frac{a}{b}$

Step 2:Applying given conditions

case 1

According to the question,

$$\begin{gathered} \frac{a+1}{b+1}=\frac{1}{2} \\ \Rightarrow 2\left(a+1\right)=b+1 \\ \Rightarrow 2a+2=b+1 \\ \Rightarrow 2a-b=1-2 \\ \Rightarrow 2a-b=-1.......................\left(1\right) \end{gathered}$$

case 2

According to the question,

$$\begin{gathered} \frac{a-5}{b-5}=\frac{5}{11} \\ \Rightarrow 11\left(a-5\right)=5(b-5) \\ \Rightarrow 11a-55=5b-25 \\ \Rightarrow 11a-5b=55-25 \\ \Rightarrow 11a-5b=30.................................\left(2\right) \end{gathered}$$

step:3

To find $a$and $b$using elimination method

$$\begin{gathered} \left(1\right)\times 5\Rightarrow 10a-5b=-5\to \left(3\right) \\ \left(2\right)\Rightarrow 11a-5b=30\to \left(2\right) \\ \left(2\right)-\left(3\right)\Rightarrow a=35 \end{gathered}$$

To get $b$ substitute $a=35$in (1)

$$\begin{gathered} \left(1\right)\Rightarrow 2\times 35-b=-1 \\ \Rightarrow 70-b=-1 \\ \Rightarrow -b=-1-70 \\ \Rightarrow -b=-71 \\ \Rightarrow b=71 \end{gathered}$$

Therefore the values of $a=35andb=71$

Checking,

$$\begin{gathered} \frac{a+1}{b+1}=\frac{35+1}{71+1}=\frac{36}{72}=\frac{1}{2} \\ \frac{a-5}{b-5}=\frac{35-5}{71-5}=\frac{30}{66}=\frac{10}{22}=\frac{5}{11} \end{gathered}$$