If AB + BA = 99, what is A + B?
Understand the Problem
The question is asking us to find the sum of two digits A and B given the equation AB + BA = 99, where AB and BA are two-digit numbers formed by digits A and B.
Answer
$A + B = 9$
Answer for screen readers
$A + B = 9$
Steps to Solve
- Represent the two-digit numbers AB and BA in terms of A and B.
The two-digit number AB can be represented as $10A + B$. Similarly, the two-digit number BA can be represented as $10B + A$.
- Substitute these expressions into the given equation.
We are given $AB + BA = 99$. Substituting the expressions from step 1, we get: $$(10A + B) + (10B + A) = 99$$
- Simplify the equation.
Combine like terms: $$11A + 11B = 99$$
- Factor out the common factor.
Factor out 11 from the left side of the equation: $$11(A + B) = 99$$
- Solve for A + B.
Divide both sides of the equation by 11: $$A + B = \frac{99}{11}$$ $$A + B = 9$$
$A + B = 9$
More Information
The sum of the digits A and B is 9. There can be multiple solutions for A and B such as A=1 and B=8 or A=2 and B=7, but the sum remains constant at 9.
Tips
A common mistake is to treat "AB" as a product of A and B instead of a two-digit number. Remember that when two digits are written together like this, it represents a number in the form $10A + B$. Another mistake would be incorrect algebraic manipulation of the original equation e.g. not correctly combining like terms.
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