Find the value of y if $402_y = 102_{ten}$

Analysis

The question asks us to find the base $y$ such that the number $402$ in base $y$ is equal to $102$ in base ten. This is a problem involving Number Bases Conversion.

Concept

To convert a number from any base $y$ to base ten, we expand the digits by multiplying each digit by the base raised to the power of its position (starting from 0 on the right).

For a number $abc_y$, the value in base ten is: $(a \times y^2) + (b \times y^1) + (c \times y^0)$

Solution

Given: $402_y = 102_{10}$

1. Expand the left side ($402_y$): $(4 \times y^2) + (0 \times y^1) + (2 \times y^0) = 102$

Since $y^0 = 1$: $4y^2 + 0 + 2 = 102$

2. Simplify the equation: $4y^2 + 2 = 102$

3. Isolate the term with $y$: Subtract 2 from both sides: $4y^2 = 102 - 2$ $4y^2 = 100$

4. Solve for $y^2$: Divide both sides by 4: $y^2 = \frac{100}{4}$ $y^2 = 25$

5. Find $y$: Take the square root of both sides: $y = \sqrt{25}$ $y = 5$

Conclusion

The value of $y$ is 5.

Exam Tip

When dealing with number bases, always remember that the base must be greater than any digit present in the number. Since the number $402$ contains the digit '4', the base $y$ must be greater than 4. This immediately allows you to eliminate options A (4), B (2), and D (3) if you were guessing, leaving 5 as the only logical choice!