Xavier and Yolanda paid their restaurant bills with $10 bill

Question

Xavier and Yolanda paid their restaurant bills with $10 bills, including tips of 20% and 24%, respectively. If each bill was for a whole number of dollars and each left the restaurant without taking any change, what’s the smallest possible amount that they paid in total?

A) $275
B) $300
C) $325
D) $340
E) $350

A) $275
B) $300
C) $325
D) $340
E) $350

MarkGMATMentor · Answer

This question is a great example of the importance of converting decimals to fractions, especially when working with a divisibility / multiples problem.

How do we know multiples are involved? Xavier and Yolanda paid in $10 bills with no change, so their payments must be multiples of 10. And we want the  least  multiple of 10 in each case.

Xavier: 
He left a 20% tip on his bill (call it x), so he left 1.2x dollars. This amount is equal to a multiple of 10:
 1.2x=10n , where both x and n are integers

Working with fractions will simplify things:
 \frac{6}{5}x=10n 
 x=\frac{50}{6}n=\frac{25}{3}n

x must be an integer, so we need to eliminate the 3 in the denominator. The smallest n that will do that is 3. Thus Xavier left 10(3) = $30.

Yolanda: 
We can follow the same process here. I'll reuse x and n for simplicity (they are not the same x and n used for Xavier).

Yolanda left a 24% tip on her bill, so:
 1.24x=10n , where both x and n are integers

Again, convert to a fraction:
 \frac{31}{25}x=10n 
 x=\frac{250}{31}n

This fraction can't be reduced, so the smallest n that will eliminate 31 is 31. Thus Yolanda left 10(31) = $310. (What did she eat?!)

Together, the smallest amount they can spend is 30+310 = $340.  The answer is D.

In most cases, without a calculator fractions are easier to work with than decimals. On divisibility problems, working with fractions is essential.

kevincan · Answer

What a generous explanation! Thank you

Xavier and Yolanda paid their restaurant bills with $10 bill

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