Question

The ratio of the rose plants to marigold plants in an orchard is 2:3. If 5 more plants of

each type are planted, the ratio of plants would be 5:7. Then find the number of rose

plants and marigold plants in the orchard.
​

Answer 1

Answer :-

$\: \: \underline{\boxed{\sf{\purple{Let's \: understand \: to \: the \: voice \: of \: question\: (Analysis)}}}}$

Here we see that we are needed to find two different values. We can find them using Linear Equations in Two Variables. Then we can make tue value of both unknown values depend on other to find them both. Let's do this question, using this concept.

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★ Question :-

The ratio of the rose plants to marigold plants in an orchard is 2:3. If 5 more plants of each type are planted, the ratio of plants would be 5:7. Then find the number of rose plants and marigold plants in the orchard.

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★ Solution :-

Given,

» Number of rose plant : Number of Marigold Plant = 2:3

» 5 + Number of Rose Plant : 5 + Number of Marigold Plant = 5 : 7

• Let the number of rose plant be 'x'.

• Let the number of Marigold plant be 'y'.

The according to the question,

~ Case I :-

$\: \: \huge{\orange{\longrightarrow \: \: \dfrac{x}{y} \: = \: \dfrac{2}{3}}}$

By cross multiplication, we get,

➔ 3x = 2y

➔ y = ½ × (3x) ....(i)

~ Case II :-

$\: \: \huge{\orange{\longrightarrow \: \: \: \dfrac{x \: + \: 5}{y \: + \: 5} \: = \: \dfrac{5}{7}}}$

By cross multiplication, we get,

➔ 7(x + 5) = 5(y + 5)

➔ 7x + 35 = 5y + 25

➔ 7x - 5y = 25 - 35

➔ 7x - 5y = -10 ...(ii)

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

↬ 7x - (5 × ½ × 3x) = -10

Multiplying all the terms by 2 , at each side, we get,

↬ 14x - 15x = -20

↬ -x = -20

Cancelling the negative sign, we get,

↬ x = 20

• Hence, the number of rose plants = x = 20.

From equation (i) and value of x, we get,

↬ y = ½ × 3x

↬ y = ½ × 3(20)

↬ y = ½ × 60 = 30

• Hence the number of Marigold plants = y = 30

$\: \: \underline{\boxed{\sf{\green{Thus, \: the \: number \: of \: rose \: plants \: are \: \underline{20} \: and \: that \: of \: marigold \: are \: \underline{30}}}}}$

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

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

~ Case I :-

✰ y = ½ (3x)

✰ 30 = ½(3(20))

✰ 30 = ½(60)

✰ 30 = 30

Clearly, LHS = RHS.

~ Case II :-

✰ 7x - 5y = -10

✰ 7(20) - 5(30) = -10

✰ 140 - 150 = -10

✰ -10 = -10

Clearly LHS = RHS

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

HENCE, VERIFIED.

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$\: \: \huge{\mapsto \: \:\underline{\sf{\red{More \: to \: know \: :-}}}}$

• Linear Equations are the equations formed using variables and constanta of single degrees.

• Polynomials are the equations formed using constant and variables of multiple degrees.

• Polynomials are of different types :-

• Linear Polynomial
• Quadratic Polynomial
• Cubic Polynomial
• Bi - Quadratic Polynomial

• Linear Equations are of three types :-

• Linear Equation In one Variable
• Linear Equation in Two Variables
• Linear Equation in Three Variables

• * Note :- This is the form of answer derived from Two Variables. The answer given above by @TheMoonlìghtPhoenix is by using one variable. Please do refer to it. Its also a easier and conventional method.

Answer 2

Answer:

Step-by-step explanation:

$\huge{\sf{Answer}}$

• Let the Ratio constant be x.
• So, here we can assume Rose as 2x.
• And marigold as 3x.

According to Question:-

$\dfrac{2x+5}{3x+5}= \dfrac{5}{7}$

Cross Multiplication:-

$\sf{7(2x+5)=5(3x+5)}$

$\sf{14x+35=15x+25}$

$\sf{14x-15x=-35+25}$

$\sf{-x=-10}$

Minus Cancels:-

$\sf{x= 10}$

Now, the ratio constants were 2x and 3x.

So,

$\sf{2x= 20}$

$\sf{3x= 30}$

So, there are 20 Roses and 30 Marigolds in the Orchard.

Things to Remember:-

• Always assume Ratio constant as any variable.
• Also, this is the single variable method. Refer to first answer for 2 variable method.
• Go line by line the question for framing equations.