All the quant - ch. 18 - ps #1 (page 270)
1. A bookshelf holds both paperback and hardcover books. the ratio of paperback books to hardcover books is 22 to 3. How many paperback books are on the shelf?
(1) the # of books on the shelf is between 202 and 247, inclusive
(2) If 18 paperback books were removed from the shelf and replaced with 18 hardcover books, the resulting ratio of paperback books to hardcover books would be 4 to 1
I understand how the first statement is sufficient and was able to guess that the second gives enough information to be sufficient as well, but do not fully understand the math behind how the second statement is sufficient.
In the explanation given I understand you have to cross multiply resulting in the new ratio being equal to P = 4H+(18)(4)-18 and the starting ratio being 3P = 22H, but how do you use both statements to solve for P and H? I plugged in 1 for H in the new ratio which makes P = 216, which is a 4:1 ratio. Is that incorrect?
GMAT Forum
2 postsPage 1 of 1
- gleesonc481
- Course Students
- Posts: 1
- Joined: Wed Oct 02, 2019 3:56 pm
- Sage Pearce-Higgins
- Forum Guests
- Posts: 1336
- Joined: Thu Apr 03, 2014 4:04 am
Re: All the quant - ch. 18 - ps #1 (page 270)
It seems that there are two good questions here about statement (2):
How can I see that statement (2) is sufficient with more clarity?
On a simple level, remember that ratios, by themselves, are simply proportions. They don't fix an actual number. However, when you add in a real quantity, it "fixes" the ratio at a certain level. So the fact that we have a real number of books in statement (2) is a clue that it's going to be sufficient to find the number of paperbacks.
On a more complicated level, the solution shows you how you can translate the statements into equations. If you have two equations and two unknowns, then you should be able to solve. (There are, however, a few exceptions to this rule that you should be aware of, see this post https://www.manhattanprep.com/gmat/foru ... ml#p134039). Of course, for DS problems, you don't have to solve the equations, simply to see that the equations can be solved.
What's the Math going on behind statement (2)?
Although you don't need to do this in a test situation, it's great practice for your review work to pick apart DS problems to understand the Math more deeply. Starting from the explanation to this problem in All the Quant, you could rearrange the first equation to make P = 22H / 3 and then substitute this into the second equation, i.e. you can replace P with "22H / 3" in the second equation to give an equation that can be solved for H. If you're unsure about this Math, then check out the Linear Equations section in the Foundations of Math book. This is called solving a system of equations by substitution. It's also called solving simultaneous equations by substitution.
Yes, this is incorrect. You may have just got lucky here. When solving equations, you're usually not free just to pick a number for a variable. Also, I don't quite understand how this makes a 4:1 ratio.
Finally, another way to deal with a situation like this one is to use the 'unknown multiplier' technique. Given that the first ratio of paperbacks : hardbacks is 22:3, then the number of paperbacks is 22x and the number of hardbacks is 3x, where x is an integer that is the multiplier of the ratio (i.e. the ratio is multiplied up by x, which could be 1,2,3, etc.). According to statement (2) we know that the new number of paperbacks is 22x – 18, and the number of hardbacks is 3x + 18, so that we can make a new equation that (22x – 18)/(3x + 18) = 4/1. This can be solved for x, so that we can then calculate the number of paperbacks.
How can I see that statement (2) is sufficient with more clarity?
On a simple level, remember that ratios, by themselves, are simply proportions. They don't fix an actual number. However, when you add in a real quantity, it "fixes" the ratio at a certain level. So the fact that we have a real number of books in statement (2) is a clue that it's going to be sufficient to find the number of paperbacks.
On a more complicated level, the solution shows you how you can translate the statements into equations. If you have two equations and two unknowns, then you should be able to solve. (There are, however, a few exceptions to this rule that you should be aware of, see this post https://www.manhattanprep.com/gmat/foru ... ml#p134039). Of course, for DS problems, you don't have to solve the equations, simply to see that the equations can be solved.
What's the Math going on behind statement (2)?
Although you don't need to do this in a test situation, it's great practice for your review work to pick apart DS problems to understand the Math more deeply. Starting from the explanation to this problem in All the Quant, you could rearrange the first equation to make P = 22H / 3 and then substitute this into the second equation, i.e. you can replace P with "22H / 3" in the second equation to give an equation that can be solved for H. If you're unsure about this Math, then check out the Linear Equations section in the Foundations of Math book. This is called solving a system of equations by substitution. It's also called solving simultaneous equations by substitution.
I plugged in 1 for H in the new ratio which makes P = 216, which is a 4:1 ratio. Is that incorrect?
Yes, this is incorrect. You may have just got lucky here. When solving equations, you're usually not free just to pick a number for a variable. Also, I don't quite understand how this makes a 4:1 ratio.
Finally, another way to deal with a situation like this one is to use the 'unknown multiplier' technique. Given that the first ratio of paperbacks : hardbacks is 22:3, then the number of paperbacks is 22x and the number of hardbacks is 3x, where x is an integer that is the multiplier of the ratio (i.e. the ratio is multiplied up by x, which could be 1,2,3, etc.). According to statement (2) we know that the new number of paperbacks is 22x – 18, and the number of hardbacks is 3x + 18, so that we can make a new equation that (22x – 18)/(3x + 18) = 4/1. This can be solved for x, so that we can then calculate the number of paperbacks.
2 posts Page 1 of 1