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Ben has a certain amount of money in $1 and $10 bills. The number of $ 07 Jan 2022, 06:45
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C
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65% (hard)

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67% (03:08) correct 33% (02:59) wrong based on 251 sessions

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Ben has a certain amount of money in $1 and $10 bills. The number of $1 bills that Ben has, when multiplied by the number of $10 bills, is equal to the total amount of money that Ben has.

If the number of $10 bills is less than 10, then which of the following is a possible value for the total number of bills (both $1 and $10 bills) that Ben can have?

(A) 14
(B) 16
(C) 18
(D) 20
(E) 24
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C
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Ben has a certain amount of money in $1 and $10 bills. The number of $ 08 Jan 2022, 08:15
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If Ben has x one-dollar bills, and t ten-dollar bills, then Ben has 10t + x dollars. We know Ben also has tx dollars. So we have this equation, which we can solve for x:

\(\\
\begin{align}\\
10t + x &= tx \\\\
10t &= tx - x \\\\
10t &= x(t - 1) \\\\
x &= \frac{10t}{t-1}\\
\end{align}\\
\)

Now x is an integer. So the fraction 10t/(t-1) needs to cancel down completely, and t-1 must be a factor of 10t. But t-1 and t are consecutive integers, so their GCD is 1, and t-1 can't cancel anything from t. So t-1 must be a factor of 10, and t-1 can equal 1, 2, 5 or 10. Since t < 10, we can rule out the largest of these, which leaves us with three possible values of t: 2, 3 or 6. If we plug those back in to the equation x + 10t = tx, we get three solutions:

t = 2, x = 20
t = 3, x = 15
t = 6, x = 12

and the possible values of t + x, which is what we're asked to find, are 18 or 22, and only 18 is among the choices.
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Ben has a certain amount of money in $1 and $10 bills. The number of $ 08 Jan 2022, 06:04
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I think it is C. The way I came to my solution is maybe not the most elegant one, but I´ll explain:

t = number of $10 bills
x = number of $1 bills

t*x must be equal to 10t + 1x

Also: t<10, so t will be between 1 and 9 (if t was 0 then Ben would have no money, since t*x = the amount of money he has)

So we have: tx = 10t + x which is t = \(\frac{10t}{x}\) + 1

Now, we can try out for every t between 1 and 9 whether this equation makes sense.

If t = 9, then 9 = \(\frac{90}{x}\) + 1 <=> 8 = \(\frac{90}{x}\)

However, there is no integer that divides 90 so that we get to 8. t=8 does not make any sense.

Repeat this until you get to t=3:

3 = \(\frac{30}{x}\) + 1 <=> 2 = \(\frac{30}{x}\) <=> x = 15

t + x = 3 + 15 = 18

So C = 18 is the right answer.

However, as I said: it´s not the most elegant way to solve this problem. I appreciate your ideas!
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Ben has a certain amount of money in $1 and $10 bills. The number of $ 06 Oct 2024, 00:41
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IanStewart
If Ben has x one-dollar bills, and t ten-dollar bills, then Ben has 10t + x dollars. We know Ben also has tx dollars. So we have this equation, which we can solve for x:

\(\\
\begin{align}\\
10t + x &= tx \\\\
10t &= tx - x \\\\
10t &= x(t - 1) \\\\
x &= \frac{10t}{t-1}\\
\end{align}\\
\)

Now x is an integer. So the fraction 10t/(t-1) needs to cancel down completely, and t-1 must be a factor of 10t. But t-1 and t are consecutive integers, so their GCD is 1, and t-1 can't cancel anything from t. So t-1 must be a factor of 10, and t-1 can equal 1, 2, 5 or 10. Since t < 10, we can rule out the largest of these, which leaves us with three possible values of t: 2, 3 or 6. If we plug those back in to the equation x + 10t = tx, we get three solutions:

t = 2, x = 20
t = 3, x = 15
t = 6, x = 12

and the possible values of t + x, which is what we're asked to find, are 18 or 22, and only 18 is among the choices.
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Hi IanStewart

In hindsight your method seems so obvious.
But, I started with an equation for t with x and after applying the constraint got x > 100/9 which threw me into a loop.
Did you make an intended effort to form the equation for x so that gcd property of consecutive numbers can be used?

Is it coming from experience or some clue in the question?
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Ben has a certain amount of money in $1 and $10 bills. The number of $ 06 Oct 2024, 13:59
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Quote:
Hi IanStewart

In hindsight your method seems so obvious.
But, I started with an equation for t with x and after applying the constraint got x > 100/9 which threw me into a loop.
Did you make an intended effort to form the equation for x so that gcd property of consecutive numbers can be used?

Is it coming from experience or some clue in the question?
Show more

When I look at a question like this, I'd think about it as follows:

• the question asks for "a possible value" of t + x. That usually means there are at least two possible answers to the question, and only one of those answers appears among the answer choices. So we will either end up with an equation like a quadratic, with two (or more) solutions, only one of which will appear among the answer choices, or more often, a situation where there are several possible answers, and where we may need to test some values. Either way, I'll want to translate to algebra, to see if we get a quadratic, or to see if we can find some way to limit the options we might need to test;

• after producing the equation 10t + x = tx, because the question asks for the value of t + x, the first thing I'd always do is to try to rewrite the equation with t + x on one side, because then I might immediately learn a lot about the quantity the question asks us to find. That doesn't turn out to help much here, as far as I can see, but is often the fastest way to do a lot of similar questions;

• since we can't learn about t + x directly, the most obvious thing to do is to try to learn something about either t or x alone. It really doesn't matter which letter you solve for -- in my solution I solved for x, but if you instead solve for t, you get this equation:

10t + x = tx
x = tx - 10t
x = t(x - 10)
t = x/(x - 10)

Notice now x is divisible by x-10; these numbers are 10 apart, so x-10 would need to be a divisor of 10, so can only be 1, 2, 5 or 10 (there are a few ways to see that -- it's maybe easier to let y = x - 10, so we know y + 10 is divisible by y, or (y + 10)/y is an integer, so 1 + (10/y) is an integer and y = x - 10 is a divisor of 10). So x can only be 11, 12, 15 or 20, and we have only a few options to investigate.

I did not focus on the constraint t < 10 (though in a different problem, perhaps I would have) because it doesn't seem very restrictive here -- even with that constraint, we have many possible values of t. Not only that, but we already know, glancing at the answer choices, that t can't be very large, so the restriction doesn't seem likely to help much. If we're going to narrow down our possibilities here, we'll have to look at the other information in the problem.
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Ben has a certain amount of money in $1 and $10 bills. The number of $ 06 Oct 2024, 19:54
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IanStewart
Quote:
Hi IanStewart

In hindsight your method seems so obvious.
But, I started with an equation for t with x and after applying the constraint got x > 100/9 which threw me into a loop.
Did you make an intended effort to form the equation for x so that gcd property of consecutive numbers can be used?

Is it coming from experience or some clue in the question?

When I look at a question like this, I'd think about it as follows:

• the question asks for "a possible value" of t + x. That usually means there are at least two possible answers to the question, and only one of those answers appears among the answer choices. So we will either end up with an equation like a quadratic, with two (or more) solutions, only one of which will appear among the answer choices, or more often, a situation where there are several possible answers, and where we may need to test some values. Either way, I'll want to translate to algebra, to see if we get a quadratic, or to see if we can find some way to limit the options we might need to test;

• after producing the equation 10t + x = tx, because the question asks for the value of t + x, the first thing I'd always do is to try to rewrite the equation with t + x on one side, because then I might immediately learn a lot about the quantity the question asks us to find. That doesn't turn out to help much here, as far as I can see, but is often the fastest way to do a lot of similar questions;

• since we can't learn about t + x directly, the most obvious thing to do is to try to learn something about either t or x alone. It really doesn't matter which letter you solve for -- in my solution I solved for x, but if you instead solve for t, you get this equation:

10t + x = tx
x = tx - 10t
x = t(x - 10)
t = x/(x - 10)

Notice now x is divisible by x-10; these numbers are 10 apart, so x-10 would need to be a divisor of 10, so can only be 1, 2, 5 or 10 (there are a few ways to see that -- it's maybe easier to let y = x - 10, so we know y + 10 is divisible by y, or (y + 10)/y is an integer, so 1 + (10/y) is an integer and y = x - 10 is a divisor of 10). So x can only be 11, 12, 15 or 20, and we have only a few options to investigate.

I did not focus on the constraint t < 10 (though in a different problem, perhaps I would have) because it doesn't seem very restrictive here -- even with that constraint, we have many possible values of t. Not only that, but we already know, glancing at the answer choices, that t can't be very large, so the restriction doesn't seem likely to help much. If we're going to narrow down our possibilities here, we'll have to look at the other information in the problem.
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Great explanation. Thank you !
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Ben has a certain amount of money in $1 and $10 bills. The number of $ 08 Oct 2024, 06:51
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IanStewart
If Ben has x one-dollar bills, and t ten-dollar bills, then Ben has 10t + x dollars. We know Ben also has tx dollars. So we have this equation, which we can solve for x:

\(\\
\begin{align}\\
10t + x &= tx \\\\
10t &= tx - x \\\\
10t &= x(t - 1) \\\\
x &= \frac{10t}{t-1}\\
\end{align}\\
\)

Now x is an integer. So the fraction 10t/(t-1) needs to cancel down completely, and t-1 must be a factor of 10t. But t-1 and t are consecutive integers, so their GCD is 1, and t-1 can't cancel anything from t. So t-1 must be a factor of 10, and t-1 can equal 1, 2, 5 or 10. Since t < 10, we can rule out the largest of these, which leaves us with three possible values of t: 2, 3 or 6. If we plug those back in to the equation x + 10t = tx, we get three solutions:

t = 2, x = 20
t = 3, x = 15
t = 6, x = 12

and the possible values of t + x, which is what we're asked to find, are 18 or 22, and only 18 is among the choices.
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This is a great solution. Do you think we can use backsolving/substitution here in any way? Just curious?
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Ben has a certain amount of money in $1 and $10 bills. The number of $ 11 Oct 2024, 07:42
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# 1 currency = a
# 10 curreny = b
Given a*b = a + 10 b & b <10

I just started plugging values starting b = 9 and reducing b by 1 each time i didn't get an integer a .
When b = 6, a =12 therefore a+b = 18..

I saw the solution posted above by Ian and upon reading through it looks pretty obvious, but that did not strike me while solving the question.
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Ben has a certain amount of money in $1 and $10 bills. The number of $ 05 Feb 2026, 12:02
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