Question

Two balls weigh $15\;kg$. If one is $4$ times heavier than the other, what is the mass of the heavier ball?
A. $15\;kg$
B. $12\;kg$
C. $9\;kg$
D. $4\;kg$

Answer 1

Hint:The basic concept of algebra and deducing the mathematical terms given in the question into mathematical equations and symbols to represent the mathematical operation that is performed leads to the required answer. A relation between the masses of the two balls needs to be constructed to find out which ball is heavier.

Complete step by step answer:
The above problem revolves around the concepts of algebra which is a branch of mathematics which governs certain rules which are to be followed when constructing a mathematical equation using basic mathematical symbols and operations such as addition, subtraction, multiplication and division.

This problem is a word problem which we are supposed to find the solution to by constructing mathematical equations in accordance with what the word problem gives as a sentence. Let us now analyze the problem. The problem asks us to find the mass of the heavier ball amongst two given balls which have a certain weight. To find this out we first need to know mass means.

Mass in simple terms in this case refers to the weight of the ball and hence we are required to find the masses of the respective balls and draw a comparison between them to determine which is heavier. Heaviness is measured in terms of the mass or the weight of the ball as the ball with a greater mass in comparison to the other will be considered as the heavier ball.

Knowing this, we will now extract the data given from the question to put it in the form of mathematical equations. The question says that there are two balls which weigh a total of $15\;kg$. This means that the masses of both the balls combined is given to be $15\;kg$. In order to construct the equations we first take the mass of one of the two balls to be denoted by a variable $x$. This variable is an unknown quantity and represents the mass of the first ball.

Let, ${m_1} = x$ ---------- ($1$)
Also, it is given that the mass of the second ball is four times the first ball. The term ‘times’ here refers to multiplication operations in mathematics and hence we have the equation relating the mass of the second ball and the first ball. We write an equation for the mass of the second ball in terms of the first ball. Thus, we have:
$ \Rightarrow {m_2} = 4{m_1}$
But we have taken the mass of the first ball as $x$. Hence, the above equation becomes:
$ \Rightarrow {m_2} = 4x$ ----------($2$)

We already know that the masses of the two balls together as a total is given as $15\;kg$ and hence we can say that the sum of the masses of the first and second ball will be equal to $15\;kg$. Therefore, when we write this in a mathematical equation to get:
${m_1} + {m_2} = 15$
From equations ($1$) and ($2$) we get:
$x + 4x = 15$
We now solve the above equation to get:
$5x = 15$
$ \Rightarrow x = \dfrac{{15}}{5}$
Hence we get:
$x = 3\;kg$

This is the mass of the first ball that we have considered. Now, we must find the mass of the second ball. We know from equation ($2$) how the masses of the two balls are related to each other hence:
We know that,
$ \Rightarrow {m_2} = 4x$
By substituting the value of $x$, that is, the mass of the first ball we get:
${m_2} = 4 \times 3$
$ \Rightarrow {m_2} = 12\;kg$
This is the mass of the second ball. Now, that we found the masses of the two balls we can compare them to find the heavier ball. Since, the second ball has a greater mass since its weight is $12\;kg$ it is the heavier ball.

Therefore, the correct option is option B.

Additional information: The quantities of mass and weight seem to be similar concept-wise but they are in reality not the same or equal in value. The mass is said to be the amount of matter contained in the body while the weight is said to be the amount of force that is applied on the body due to the effect of gravity.

Note:The common error that can be performed in these types of problems is in framing the equations relating the two masses of the balls. The common misconception here is that the second ball is four times heavier than the first ball itself so it is related directly to the mass of the first ball and hence the equation for mass of second ball should be in terms of the first ball’s mass only.