Question

Two numbers are in the ratio of 5:8.If 4 is subtracted from each then the ratio become 1:2. The small number is
O 16
0 4
O 10
O 2
O None of these​

Answer 1

Given :

Two numbers are in the ratio of 5:8.

If 4 is subtracted from each then the ratio become 1:2.

To find :

The smaller number .

Solution :

As the numbers are in ratio of 5 : 8, so :

Let the numbers be 5m and 8m respectively.

Now atq,

⇒ (5m - 4) : (8m - 4) = 1 : 2

⇒ (5m - 4)/(8m - 4) = 1/2

⇒ 8m - 4 = 2(5m - 4)

⇒ 8m - 4 = 10m - 8

⇒ 10m - 8m = -4 + 8

⇒ 2m = 4

⇒ m = 4/2

⇒ m = 2

Now the numbers :

Smaller number = 5m = 5(2) = 10

Bigger number = 8(2) = 16

Therefore,

Smaller number = 10  (Option : C)

Answer 2

Answer :-

$\: \: \boxed{\boxed{\rm{\mapsto \: \: \: Firstly \: let's \: understand \: the \: concept \: used}}}$

Here the concept of Linear Equations in Two Variables has been used. According to this if we are given unknown quantities then we can find them using the constant terms. Here we are going to take both the numbers are variables.

Let's do it !!

_______________________________________________

★ Question :-

Two numbers are in the ratio of 5:8.If 4 is subtracted from each then the ratio become 1:2. The small number is ?

_______________________________________________

★ Solution :-

Given,

» Initial ratio of two numbers = 5 : 8

» If 4 is subtracted from each number, then the ratio becomes = 1 : 2

• Let the first number be 'x' and the second number be 'y'. Then,

$\: \tt{\leadsto \: \: Ratio \: of \: two \: numbers \: = \: \large{\dfrac{x}{y}}}$

Now according to the question :-

~ Case I :-

$\: \qquad \large{\bf{\Longrightarrow \: \: \dfrac{x}{y} \: = \: \dfrac{5}{8}}}$

By cross multiplication, we get,

➣ 8x = 5y

➣ x = ⅝ × y ... (i)

~ Case II :-

$\: \qquad \large{\bf{\Longrightarrow \: \: \dfrac{x \: - \: 4}{y \: - \: 4} \: = \: \dfrac{1}{2}}}$

By cross multiplication, we get,

➣ 2(x - 4) = 1(y - 4)

➣ 2x - 8 = y - 4

➣ 2x - y = - 4 + 8

➣ 2x - y = 4 ... (ii)

From equation, (i) and (ii), we get,

➺ 2(⅝ × y) - y = 4

$\: \\ \large{\bf{\Longrightarrow \: \: \dfrac{10y}{8} \: - \: y \: = \: 4}}$

Multiplying all the terms by 8, we get,

➺ 10y - 8y = 32

➺ 2y = 32

$\: \large{\rm{\longrightarrow \: \: \qquad y \: = \: \dfrac{32}{2} \: = \: 16}}$

$\: \qquad \huge{\boxed{\sf{y \: = \: 16}}}$

By using equation (i) and the value of y, we get

➺ x = ⅝ × y

➺ x = ⅝ × 16

➺ x = 5 × 2 = 10

$\: \qquad \huge{\boxed{\sf{x \: = \: 10}}}$

So, x = 10 and y = 16. Hence, we can get our answer as :-

• Hence, the smaller number = 10 (Option C)

$\\ \large{\boxed{\leadsto{\rm{Thus, \: the \: smaller \: number \: is \: \boxed{10}}}}}$

_______________________________________________

$\: \: \underline{\underline{\rm{\Longrightarrow \: \: Confused? \: Don't \: worry \: let's \: verify \: it \: :-}}}$

For verification, we need to simply apply the values we got into the equations we formed, then,

~ Case I :-

✒ x = ⅝ × y

✒ x = ⅝ × 16

✒ 10 = 5 × 2 = 10

Clearly, LHS = RHS

~ Case II :-

✒ 2(x - 4) = 1(y - 4)

✒ 2(10 - 4) = 1(16 - 4)

✒ 2(6) = 12

✒ 12 = 12

Clearly, LHS = RHS

Here both the conditions satisfies. So our answer is correct.

Hence, Verified.

_______________________________________________

$\: \: \: \huge{\boxed{\tt{\large{More \: to \: know \: :-}}}}$

• Polynomials are the equations formed using constant and variable terms but of many degrees.

• Linear Equations are the equations formed using constant and variable terms but of single degrees.