CAT 2025 Lesson : Linear Equations - Basics of Equations

Vijay scored $4$ marks more than twice the marks that Antony scored. Tasneem scored $8$ marks more than half the marks scored by Vijay. If Antony scored $45$ marks, then how many marks did Tasneem score?

Solution

Antony scored $45$ marks. And, Vijay's marks is $4$ more than twice that of Antony's

$\therefore$ Vijay's marks $= 4 + (2 \times 45) = 94$

Tasneem's marks is $8$ more than half of Vijay's marks.

$\therefore$ Tasneem's marks $= 8 + \dfrac{94}{2} = 8 + 47 = 55$

Answer: $55$

A magician was given a certain number of coins. She doubled the number of coins and gave $8$ coins to John. She then doubled the remaining coins and gave $8$ coins to Doe. She was now left with no coins. How many coins were given to the magician?

Solution

Identifying Variables

We need to find the number of coins initially given to the magician. Let this be denoted by the variable $\bm{n}$.

Note: In most questions, the quantity/amount to be found will be the variable(s) of the equation(s).

Forming Equations

Each statement in the question is provided in quotes below.

“A magician was given a certain number of coins.” ⇒ Coins with magician $= \bm{n}$

“She doubled the number of coins and gave $8$ coins to John.” ⇒ Coins with magician $= \bm{2n - 8}$

“She then doubled the remaining coins and gave 8 coins to Doe.” ⇒ Coins with magician $= \bm{2(2n - 8) - 8}$

“She was now left with no coins.” ⇒ $\bm{2(2n - 8) - 8 = 0}$

Solving Equations (with $\bm{1}$ variable)

Expand the equation to remove all brackets.
$2(2n - 8) - 8 = 0$
⇒ $4n - 16 - 8 = 0$
⇒ $\bm{4n - 24 = 0}$

Keep the variable on one side and take the constant to the other.
⇒ $\bm{4n = 24}$

Take the coefficient of the variable to the other side.
⇒ $n = \dfrac{24}{4}$ ⇒ $\bm{n = 6}$

Alternatively

The above explanation explains the basics of forming equations in detail. Going forward, you need to reduce the steps as follows.

Coins with Magician	Number of coins
initially	$n$
after doubling and giving 8 to John	$2n - 8$
after doubling and giving 8 to Doe	$2(2n – 8) – 8 = 4n – 24$

As the magician is now left with no coins, $4n - 24 = 0$
⇒ $4n = 24$
⇒ $n = \dfrac{24}{4} = 6$ coins

Answer: $6$ coins

A trader sold $5$ cows and $4$ sheep for Rs. $22,000$, while he sold $3$ cows and $2$ sheep for Rs. $12,000$. How much is $1$ cow sold for?

Solution

Identifying Variables
In this question, we need to find the price of a cow. The question provides the price of cows and sheep sold together.

Let the variables $\bm{c}$ and $\bm{s}$ denote price of $\bm{1}$ cow and $\bm{1}$ sheep respectively.

Forming Equations

$5$ cows and $4$ sheep are sold for Rs. $22,000$
$5c + 4s = 22000$ -----($1$)

$3$ cows and $2$ sheep are sold for Rs. $12,000$
$3c + 2s = 12000$ -----($2$)

We need to multiply one or both the equations by a constant so that the coefficients of one variable is the same in both equations.

$2$ multiplied by equation ($2$) results in the same coefficient for $s$ and becomes $6c + 4s = 24000$ --- ($3$)

From this point the equations can be solved in $2$ ways:

$\bm{1}$) Subtraction Method

Subtract ($1$) from ($3$)

$6c + 4s = 24000$
$5c + 4s = 22000$
($-$)  ($-$)    ($-$)
------------------------
$1c = 2000$

So, $1$ cow was sold for Rs. $2000$

$\bm{2}$) Substitution Method

Express a variable from an equation in terms of the other variable and substitute it in the other equation.

In equation ($1$), derive the equation for $s$ as follows
$5c + 4s = 22000$
⇒ $4s = 22000 - 5c$

⇒ $s = \dfrac{22000 - 5c}{4}$

Now substitute this value of $s$ in ($2$).

($2$) becomes ⇒ $3c + 2 \times \left(\dfrac{22000 - 5c}{4}\right) = 12000$

⇒ $3c + \dfrac{22000 - 5c}{2} = 12000$

⇒ $\dfrac{6c + 22000 - 5c}{2} = 12000$

⇒ $c + 22000 = 24000$
⇒ $c = 2000$

Answer: Rs $2000$