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# Sandra is paid an hourly base rate for babysitting up to two children

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Sandra is paid an hourly base rate for babysitting up to two children  [#permalink]

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05 Nov 2018, 11:32
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Difficulty:

95% (hard)

Question Stats:

34% (02:26) correct 66% (02:26) wrong based on 61 sessions

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Sandra is paid an hourly base rate for babysitting up to two children a day, and an additional 20% per hour for each additional child. If her hours are rounded up to the next full hour, how many hours was Sandra paid for last week?

1. One day last week, Sandra was paid \$51 to babysit 3 children for 5 hours.

2. Sandra was paid a total of \$108.80 last week and never babysat more than 3 children at once.

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Re: Sandra is paid an hourly base rate for babysitting up to two children  [#permalink]

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05 Nov 2018, 14:14
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1
AkshdeepS wrote:
Sandra is paid an hourly base rate for babysitting up to two children a day, and an additional 20% per hour for each additional child. If her hours are rounded up to the next full hour, how many hours was Sandra paid for last week?

1. One day last week, Sandra was paid \$51 to babysit 3 children for 5 hours.

2. Sandra was paid a total of \$108.80 last week and never babysat more than 3 children at once.

Question stem: Let's say that her rate is R dollars an hour for 1 or 2 children, and 1.2R for 3 children. (Also, 1.44R for 4 children, although I didn't end up using that in my solution.)

We don't know how much money she was paid, and we don't know how many children she babysat.

We want to know how many hours she worked for.

Statement 1: Since this only tells us info about a single day, it's insufficient. However, note that this would tell us her rate (the value of R).

Statement 2: Quickly test cases. She could have babysat 1 child for 1 hour at \$108.80 per hour, or 1 child for 2 hours at half that rate. Since those are two different numbers of hours (1 versus 2), this statement is insufficient.

Statement 1 + 2: Now, things get interesting!

Her rate for 3 children for 5 hours was \$51. So,

5*1.2R = 51
6R = 51
R = 8.5

She normally makes \$8.50 per hour. If she has three children, she makes \$10.20 per hour.

She earned \$108.80 for a round number of hours. She never had more than 3 children during those hours. We could write this as an equation:

108.8 = 8.5(hours with 1 or 2 children) + 10.2(hours with 3 children)

The question becomes, is there only one way to solve this? Or is there more than one way?

What I noticed was that the digit after the decimal on the left is an 8. On the right, we have a .5 and a .2. Adding a .5 will make the tenths digit odd, so we need to have an even number of hours at the 8.5 rate to cancel that out. The .5 will turn into a .0.

Since the 8.5 doesn't contribute anything to the tenths digit, we need to focus on the 10.2. The .2 needs to turn into a .8. How can that happen? Only if there are certain numbers of hours:

4 hours = 4(10.2) = 40.8
9 hours = 9(10.2) = 91.8

We can stop there, since if there are any more hours, we'll end up with more than \$108.80 in total.

If there are 9 hours with three children, the remaining amount of money is 108.8 - 91.8, or 17.0. \$17.0 is what she'd make in exactly two hours with one or two children. So, the total number of hours is 9 + 2 = 11.

If there are 4 hours with three children, the remaining amount of money is 108.8 - 40.8, or 68.0. \$68.0 is what she'd make in exactly eight hours with one or two children. So, the total number of hours is 4 + 8 = 12.

HOWEVER, There can't be 4 hours with 3 children, since the first statement tells us that there were at least 5 hours with 3 children in total! The only possible case is the one with 9 hours with 3 children, and 2 hours with 1 or 2 children. 11 hours in total.

The answer is 11, so the answer to the DS problem is C, both statements together are sufficient.
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Re: Sandra is paid an hourly base rate for babysitting up to two children  [#permalink]

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11 Apr 2019, 01:52
ccooley wrote:
AkshdeepS wrote:
Sandra is paid an hourly base rate for babysitting up to two children a day, and an additional 20% per hour for each additional child. If her hours are rounded up to the next full hour, how many hours was Sandra paid for last week?

1. One day last week, Sandra was paid \$51 to babysit 3 children for 5 hours.

2. Sandra was paid a total of \$108.80 last week and never babysat more than 3 children at once.

Question stem: Let's say that her rate is R dollars an hour for 1 or 2 children, and 1.2R for 3 children. (Also, 1.44R for 4 children, although I didn't end up using that in my solution.)

We don't know how much money she was paid, and we don't know how many children she babysat.

We want to know how many hours she worked for.

Statement 1: Since this only tells us info about a single day, it's insufficient. However, note that this would tell us her rate (the value of R).

Statement 2: Quickly test cases. She could have babysat 1 child for 1 hour at \$108.80 per hour, or 1 child for 2 hours at half that rate. Since those are two different numbers of hours (1 versus 2), this statement is insufficient.

Statement 1 + 2: Now, things get interesting!

Her rate for 3 children for 5 hours was \$51. So,

5*1.2R = 51
6R = 51
R = 8.5

She normally makes \$8.50 per hour. If she has three children, she makes \$10.20 per hour.

She earned \$108.80 for a round number of hours. She never had more than 3 children during those hours. We could write this as an equation:

108.8 = 8.5(hours with 1 or 2 children) + 10.2(hours with 3 children)

The question becomes, is there only one way to solve this? Or is there more than one way?

What I noticed was that the digit after the decimal on the left is an 8. On the right, we have a .5 and a .2. Adding a .5 will make the tenths digit odd, so we need to have an even number of hours at the 8.5 rate to cancel that out. The .5 will turn into a .0.

Since the 8.5 doesn't contribute anything to the tenths digit, we need to focus on the 10.2. The .2 needs to turn into a .8. How can that happen? Only if there are certain numbers of hours:

4 hours = 4(10.2) = 40.8
9 hours = 9(10.2) = 91.8

We can stop there, since if there are any more hours, we'll end up with more than \$108.80 in total.

If there are 9 hours with three children, the remaining amount of money is 108.8 - 91.8, or 17.0. \$17.0 is what she'd make in exactly two hours with one or two children. So, the total number of hours is 9 + 2 = 11.

If there are 4 hours with three children, the remaining amount of money is 108.8 - 40.8, or 68.0. \$68.0 is what she'd make in exactly eight hours with one or two children. So, the total number of hours is 4 + 8 = 12.

HOWEVER, There can't be 4 hours with 3 children, since the first statement tells us that there were at least 5 hours with 3 children in total! The only possible case is the one with 9 hours with 3 children, and 2 hours with 1 or 2 children. 11 hours in total.

The answer is 11, so the answer to the DS problem is C, both statements together are sufficient.

This question took me around 5mins to solve. What do you suggest me to do if this type of question comes on the exam day?
Re: Sandra is paid an hourly base rate for babysitting up to two children   [#permalink] 11 Apr 2019, 01:52
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