Question

Two numbers are in the ratio \[8:3\]. If the sum of the numbers is 143, find the numbers.

Answer 1

Hint: We are provided with the ratio of two numbers and their sum to find the numbers. We will assume the common factor of both the numbers to be \[x\]. We will then form the equation based on the given conditions and then solve them to obtain the common factor. Using the common factor we will find the required numbers.

Complete step-by-step answer:
Let us assume that the common factor of both our numbers is \[x\].
Now we are given that the two numbers are in the ratio of \[8:3\]. This means that after we divide both the numbers with each other and cancel out the common factor \[x\]from both the terms we get 8 in numerator and denominator. In other words, when we divide the numerator and denominator both with \[x\], we get 8 in the numerator and 3 in the denominator.
Let us find the first number now,
From the above concept, we can see that the first number would be the product of 8 and \[x\].
Thus, the first number \[ = 8x\]
Now let us find the second number,
From the above concept, we know that the second number would be the product of 3 and \[x\].
Thus, the second number \[ = 3x\]
We are given in the question that the sum of both our numbers is 143. Forming an equation based on this information, we get
\[8x + 3x = 143\]
Adding the like terms, we get
\[ \Rightarrow 11x = 143\]
Dividing both side by 11, we get
\[\begin{array}{l} \Rightarrow \dfrac{{11x}}{{11}} = \dfrac{{143}}{{11}}\\ \Rightarrow x = 13\end{array}\]
Thus, the common factor for both of our numbers is 13.
Now we will calculate both the numbers using the common factor.
The first number \[ = 8x = 8 \times 13 = 104\]
Thus, our first number is 104.
So, our second number will be the product of 3 and the common factor 13
The second number \[ = 3x = 3 \times 13 = 39\]
Thus, our second number is 39.
\[\therefore\] The numbers are 104, and 39.

Note: The alternate way to solve this question can be as follows –
Let us assume that the two numbers are \[x\] and \[y\] with \[x > y\].
Now we are given that the ratio of the two numbers is 8 : 3. This means that,
\[\dfrac{x}{y} = \dfrac{8}{3}\]
On solving the above equation, we get
\[3x = 8y\]
\[ \Rightarrow 3x - 8y = 0\]………..\[\left( 1 \right)\]
We are also given that the sum of the two numbers is 143.
\[x + y = 143\]……………….\[\left( 2 \right)\]
We will now solve equations \[\left( 1 \right)\] and \[\left( 2 \right)\] using the method of elimination.
Multiplying equation \[\left( 2 \right)\]by 8, we get,
\[8x + 8y = 143 \times 8\]
\[ \Rightarrow 8x + 8y = 1144\]…………\[\left( 3 \right)\]
Adding equation \[\left( 1 \right)\] and \[\left( 3 \right)\], we get
\[\begin{array}{l}3x - 8y + 8x + 8y = 0 + 1144\\ \Rightarrow 11x = 1144\end{array}\]
On dividing both sides of the equation by 11, we get
\[\begin{array}{l}\dfrac{{11x}}{{11}} = \dfrac{{1144}}{{11}}\\ \Rightarrow x = 104\end{array}\]
Thus, one number is 104. Let us back-substitute the value of \[x = 104\] in equation \[\left( 2 \right)\] and then solve it. On doing so we get,
\[\begin{array}{l}104 + y = 143\\ \Rightarrow y = 143 - 104\\ \Rightarrow y = 39\end{array}\]
Hence, another number is 39.
So, our two numbers are 104 and 39.