Preet is 6 years older than Abdul. Six years ago, Preet's age was four times Abdul's age. Find their present ages.

Understand the Problem

We need to determine the current ages of Preet and Abdul based on two pieces of information: Preet is currently 6 years older than Abdul, and six years ago, Preet's age was four times Abdul's age at that time. This can be solved using a system of equations.

Answer

Preet is 14 years old and Abdul is 8 years old.

Steps to Solve

Define variables

Let $P$ be Preet's current age and $A$ be Abdul's current age.

Write the first equation

Preet is 6 years older than Abdul, so we have the equation:

$$P = A + 6$$

Write the second equation

Six years ago, Preet's age was $P - 6$ and Abdul's age was $A - 6$. Preet's age was four times Abdul's age at that time, so we have the equation:

$$P - 6 = 4(A - 6)$$

Simplify the second equation

Expand and simplify the second equation:

$$P - 6 = 4A - 24$$

$$P = 4A - 18$$

Solve the system of equations

Since $P = A + 6$ and $P = 4A - 18$, we can set the two expressions for $P$ equal to each other:

$$A + 6 = 4A - 18$$

Solve for $A$

Subtract $A$ from both sides:

$$6 = 3A - 18$$

Add 18 to both sides:

$$24 = 3A$$

Divide by 3:

$$A = 8$$

Solve for $P$

Substitute the value of $A$ into the first equation $P = A + 6$:

$$P = 8 + 6$$

$$P = 14$$

State the present ages

Preet's current age is 14 years old and Abdul's current age is 8 years old.

Preet is 14 years old and Abdul is 8 years old.

More Information

We can check our answer. Preet is indeed 6 years older than Abdul ($14 = 8 + 6$). Six years ago, Preet was $14 - 6 = 8$ years old and Abdul was $8 - 6 = 2$ years old. Preet's age (8) was four times Abdul's age (2) at that time ($8 = 4 \times 2$).

Tips

A common mistake is setting up the equations incorrectly, particularly the second equation involving their ages six years ago. Students might forget to subtract 6 from both Preet's and Abdul's current ages before applying the four times relationship. They might write $P = 4A - 6$ or $P - 6 = 4A$ which are incorrect. Another common mistake is an arithmetic error when solving the system of equations.

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