Question

Three times the larger of two numbers is equal to four times the smaller. The sum of the number is \[21\]. How do you find the numbers?

Answer 1

Hint: In this question, we have to find out two numbers. Let us assume that the smaller number is \[x\] and the larger number is \[y\]. Given, three times the larger of two numbers is equal to four times the smaller i.e., \[4x = 3y\] and the sum of the number is \[21\] i.e., \[x + y = 21\]. Solving these two linear equations will give the required numbers.

Complete step by step answer:
In this question, we have to find out two numbers. Let the smaller number be \[x\] and the larger number be \[y\]. Now, we have two variables. So, we will form two linear equations according to the condition given in the question. According to the question, three times the larger of two numbers is equal to four times the smaller. So, we can write \[4x = 3y\] i.e.,
\[x = \dfrac{3}{4}y - - - (1)\]
Also given that the sum of the number is \[21\]. So, we can write \[x + y = 21 - - - (2)\].
Putting \[(1)\] in \[(2)\], we get
\[ \Rightarrow \dfrac{3}{4}y + y = 21\]

Multiplying both the sides by \[4\], we get
\[ \Rightarrow 3y + 4y = 21 \times 4\]
On simplification, we get
\[ \Rightarrow 7y = 84\]
Dividing both the sides by \[7\], we get
\[ \Rightarrow y = 12\]
Putting the value of \[y\] in \[(1)\], we get
\[ \Rightarrow x = \dfrac{3}{4} \times 12\]
On simplification, we get
\[ \therefore x = 9\]

Therefore, the larger number is \[12\] and the smaller number is \[9\].

Note: This question is solved by solving two linear equations in which two variables are involved. A mathematical statement that has an equal to the symbol i.e., \[' = '\] is called an equation. Linear equations are equations of degree one. Also note, when any linear equation is plotted on a graph, it will necessarily produce a straight line.