Pre-Algebra

Find the base for which the following statement is true. 35 + 25 = 61

Step-by-step solution with explanation

Final Answer

Base

Step-by-step solution

1

Understand what base means here

In base , a two-digit number like 35 means . We rewrite each number using as the base, then set up an equation.
2

Write out the full equation

We expand 35, 25, and 61 in base and set the left side equal to the right side.
3

Combine like terms on the left

Add the terms together: . Add the constants: .
4

Solve for b

Subtract from both sides and subtract 1 from both sides. We get .
5

Verify the digit rule and check answer

All digits (3, 5, 2, 6, 1) must be less than the base 9 — they are. Converting both sides to base 10 gives 55 = 55. ✓

Understanding this problem

Learning Insight

Place value works the same in any base — each digit is multiplied by a power of the base. Base 10 uses powers of 10; base 9 uses powers of 9. Changing the base changes the value of every digit position, which is why the same symbols can represent a true equation in a different base.

Quick Tip

To find the base, just translate each digit-group into algebra (digit × b + digit), set up the equation, and solve for . Always double-check that every digit in the problem is strictly less than .

Common Mistake

Students forget to check that all digits must be less than the base. If the problem contained a digit equal to or greater than , that base would be invalid — always do this check at the end.