Question

1. A number is divided into two parts such that one part is 15 more than the other. If
the two parts are in the ratio 7: 4. Find the number.​

Answer 1

Answer :-

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

Here the concept or Linear Equations has been used. We know that, if we are given to find two unknown quantities which are depended on constants, we can find them by different methods of solving. Here we will use Substitution Method.

Let's do it !!

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★ Formula Used :-

$\: \\ \large{\boxed{\boxed{\sf{y_{(larger \: number)} \: = \: \bf{x_{(smaller\:number)} \: + \: 15}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{\dfrac{\: \: \: y_{(larger \: number)}}{\: \: \: x_{(smaller\: number)}} \: \: = \: \bf{\dfrac{7}{4}}}}}}$

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★ Question :-

A number is divided into two parts such that one part is 15 more than the other. If

the two parts are in the ratio 7: 4. Find the number.

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★ Solution :-

Given,

» One number is > than other number by = 15

» Ratio of two numbers = 7 : 4

Here we see thag both the numbers aren't equal, that means one is greater than the other.

So,

• Let the greater number be y

• Let the smaller number be x

Clearly, 7 denotes the ratio of y since its bigger and 4 denotes the ratio part of x.

Then according to the question,

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~ Case I :-

$\: \\ \large{\sf{:\longrightarrow \: \:\:y_{(larger \: number)} \: = \: \bf{x_{(smaller\:number)} \: + \: 15}}}$

➣ y = x + 15 ... (i)

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~ Case II :-

$\: \\ \large{\sf{:\longrightarrow \: \: \: \dfrac{ \: \: \: y_{(larger \: number)}}{\: \: \: x_{(smaller\: number)}} \: \: = \: \bf{\dfrac{7}{4}}}}$

By cross multiplication, we get,

➣ 4y = 7x .. (ii)

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

➣ 4(x + 15) = 7x

➣ 4x + 60 = 7x

➣ 7x - 4x = 60

➣ 3x = 60

$\: \\\qquad \large{\sf{:\Longrightarrow \: \: \: x \: \: = \: \: \dfrac{\cancel{60}}{\cancel{3}} \: \: = \: \: \underline{\underline{20}}}}$

$\: \\ \large{\boxed{\boxed{\tt{Hence,\: \: x \: \: = \: \: \bf{20}}}}}$

Using the value of x and equation (i), we get,

➣ y = x + 15

➣ y = 20 + 15

➣ y = 35

$\: \\ \large{\boxed{\boxed{\tt{Hence,\: \: y \: \: = \: \: \bf{35}}}}}$

Thus both numbers are 20 and 35. Now we know that the number was divided into two parts, so of course the part will be 20 and 35.

Let's add both to find the main number.

• 20 + 35 = 55

$\: \: \\ \large{\underline{\underline{\rm{\mapsto \;\;\: Thus, \; the \; numbers \; is \;\; \boxed{\bf{55}}}}}}$

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$\: \\ \large{\underline{\sf{\leadsto \: \: Confused? \; Don't \; worry \; let's \; verify \; it \; :-}}}$

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

~ Case I :-

$\: \\ \sf{:\longrightarrow \: \: \: \:\:y_{(larger \: number)} \: = \: \bf{x_{(smaller\:number)} \: + \: 15}}$

$\: \\ \sf{:\longrightarrow \: \: \: \:\:35 \: = \: \sf{20 \;\: + \;\: 15}}$

$\: \\ \sf{:\longrightarrow \: \: \: \:\:35 \: = \: \sf{35}}$

Clearly, LHS = RHS.

~ Case II :-

$\: \\ \sf{:\longrightarrow \: \: \: \dfrac{ \: \: \: y_{(larger \: number)}}{\: \: \: x_{(smaller\: number)}} \: \: = \: \bf{\dfrac{7}{4}}}$

$\: \\ \sf{:\longrightarrow \: \: \: \dfrac{ \: \: \: \cancel{35}}{\: \: \: \cancel{20}} \: \: = \: \sf{\dfrac{7}{4}}}$

Cancelling the numerator and denominator at LHS by 5, we get,

$\: \\ \sf{:\longrightarrow \: \: \: \dfrac{ \: \: \: 7}{\: \: \: 4} \: \: = \:\sf{\dfrac{7}{4}}}$

Clearly, LHS = RHS.

Here both the conditions satisfy, so our answer is correct.

Hence, Verified.

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$\: \: \\ \large{\underbrace{\underbrace{\sf{More \: \; to \: \; know \: \; :-}}}}$

• Polynomials are the mathematical expressions formed using constant and variable terms where the variable term can be of any degree.

• Linear Equations are the equations formed using constant and variable terms but the variable terms will be of single degree only that is 1.

Answer 2

Solution :

Number = 55

Step by step explanation:

Given : Ratio of two parts 7:4

Let the first part be 7 x and other be 4x

According to the question

one part is 15 more than the other.

Thus

$\sf\:7x=4x+15$

$\sf\implies\:7x-4x=15$

$\sf\implies\:3x=15$

$\sf\implies\:x=\dfrac{15}{3}$

$\sf\implies\:x=5$

Hence,

One part , 7x = 7×5=35

Other part ,4x =4×5=20

Therefore,

Number = 7x+4x =11x =55

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Verification :

Ratio of two parts is 7:4

Part one = 35

other one = 20

Then, ratio = 7:4

Hence , verified