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Difficulty:
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Question Stats:
65% (02:11) correct
35%
(02:17)
wrong
based on 398
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The value of \(\frac{(.4444)×(.3750)×(.4000)}{(.6667)×(.1667)×(.7500)}\) is closest to
A) .6
B) .80
C) 1.20
D) 1.25
E) 1.60
A) .6
B) .80
C) 1.20
D) 1.25
E) 1.60
29 Apr 2020, 10:12
Kudos
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I had a student from one of my classes pose this question, so I thought I would answer it here so that everyone could benefit from the explanation. This is a classic question that highlights fundamental strategies you need to understand as you prepare for the GMAT. These strategies are useful for whole categories of questions, not just this one. Ready? Here is the full "GMAT Jujitsu" for this question...
First, lets talk about what the GMAT is doing with this problem to trip up novice test takers. The amount of math that it would take to actually solve this the long way around is staggering. You should be able to see what I mean: just imagine multiplying out \((.4444)×(.3750)×(.4000)\) by hand and then doing long division three times with similarly messy decimals. There MUST be a quicker way to solve this problem than decimal multiplication and decimal division.
A common strategy for questions like this is Approximation, but that only works if the answer choices are far apart from each other. Just looking at the \(\pm 0.05\) difference between "C" and "D" helps us to realize Approximation isn't a good strategy, especially since multiplying and dividing approximated numbers magnifies the numerical "wiggle" associated with rounding numbers. To prove my point, let's watch what happens when we round all the messy decimals to a single digit:
\(\frac{(.4444)×(.3750)×(.4000)}{(.6667)×(.1667)×(.7500)}\approx{\frac{(.4)×(.4)×(.4)}{(.7)×(.2)×(.7)}}=\frac{4×4×4}{7×2×7}=\frac{64}{98}\approx{0.65}\)
The closest answer to \(0.65\) is answer choice "A" -- the wrong answer.
One of the powerful strategies that unlocks problems like this is what I like to call "Fractions Are Your Friends." On the GMAT, decimals are often deliberately messy, but if you see an easy way to convert a decimal into its fractional equivalent, the math goes very smoothly. I have attached a copy of a fraction-to-decimal conversion table to this response (see below). The numbers across the top of the table are the numerators of the fraction, while the numbers along the left side of the table are the denominators of the fraction. Using this table, we have the following conversions. While they are still technically approximations, they will be MUCH closer to the actual value.
\(.4444\approx{\frac{4}{9}\hspace{35pt}.375={\frac{3}{8}\hspace{35pt}.4000={\frac{2}{5}\hspace{35pt}.6667\approx{\frac{2}{3}\hspace{35pt}.1667\approx{\frac{1}{6}\hspace{35pt}.7500={\frac{3}{4}\)
Once we plug these values into the original and flip the fractions on the denominator (since dividing by a fraction is the same thing as multiplying by the inverse of that fraction), the math turns out to be pretty simple:
\(\frac{(.4444)×(.3750)×(.4000)}{(.6667)×(.1667)×(.7500)}\approx{\frac{4}{9}×\frac{3}{8}×\frac{2}{5}×\frac{3}{2}×\frac{6}{1}×\frac{4}{3}}=\frac{4×3×2×3×6×4}{9×8×5×2×1×3}\)
Now, don't multiply these numbers out. You are just going to get another enormous fraction. Instead, look for common factors in the top and bottom of the fraction and cancel them. This is a strategy I call "Divide and Conquer" in my classes. Thus,
\(\require{cancel}\frac{4×3×2×3×6×4}{9×8×5×2×1×3}=\frac{\cancel{4}×\cancel{3}×\cancel{2}×\cancel{3}×\cancel{6}×4}{\cancel{9}×\cancel{8}×5×\cancel{2}×\cancel{1}×\cancel{3}}=\frac{4}{5}\)
Using our fractions-to-decimals conversion table, we can convert the \(\frac{4}{5}\) back into its decimal form, \(0.8\). The answer is "B".
Now, let's look back on this problem through the lens of strategy. As you study for the GMAT, you don't want to just look at practice questions from the perspective of "How do I solve this problem?" Instead, use each problem to teach you strategic ways of thinking about entire categories of questions, using the leverage and structure of the problem to train your GMAT reflexes. On this question, we see three primary strategies. One of them is what I call "Mathugliness" -- the way the GMAT deliberately makes questions look harder than they are by giving you ugly math. When you see something like that, don't panic. Party. There has to be an alternative way of looking at the math or simplifying your approach. In this case, there is: convert decimals to fractions ("Fractions Are Your Friends") and then cancel out factors that are common to both numerators and denominators ("Divide and Conquer").
First, lets talk about what the GMAT is doing with this problem to trip up novice test takers. The amount of math that it would take to actually solve this the long way around is staggering. You should be able to see what I mean: just imagine multiplying out \((.4444)×(.3750)×(.4000)\) by hand and then doing long division three times with similarly messy decimals. There MUST be a quicker way to solve this problem than decimal multiplication and decimal division.
A common strategy for questions like this is Approximation, but that only works if the answer choices are far apart from each other. Just looking at the \(\pm 0.05\) difference between "C" and "D" helps us to realize Approximation isn't a good strategy, especially since multiplying and dividing approximated numbers magnifies the numerical "wiggle" associated with rounding numbers. To prove my point, let's watch what happens when we round all the messy decimals to a single digit:
\(\frac{(.4444)×(.3750)×(.4000)}{(.6667)×(.1667)×(.7500)}\approx{\frac{(.4)×(.4)×(.4)}{(.7)×(.2)×(.7)}}=\frac{4×4×4}{7×2×7}=\frac{64}{98}\approx{0.65}\)
The closest answer to \(0.65\) is answer choice "A" -- the wrong answer.
One of the powerful strategies that unlocks problems like this is what I like to call "Fractions Are Your Friends." On the GMAT, decimals are often deliberately messy, but if you see an easy way to convert a decimal into its fractional equivalent, the math goes very smoothly. I have attached a copy of a fraction-to-decimal conversion table to this response (see below). The numbers across the top of the table are the numerators of the fraction, while the numbers along the left side of the table are the denominators of the fraction. Using this table, we have the following conversions. While they are still technically approximations, they will be MUCH closer to the actual value.
\(.4444\approx{\frac{4}{9}\hspace{35pt}.375={\frac{3}{8}\hspace{35pt}.4000={\frac{2}{5}\hspace{35pt}.6667\approx{\frac{2}{3}\hspace{35pt}.1667\approx{\frac{1}{6}\hspace{35pt}.7500={\frac{3}{4}\)
Once we plug these values into the original and flip the fractions on the denominator (since dividing by a fraction is the same thing as multiplying by the inverse of that fraction), the math turns out to be pretty simple:
\(\frac{(.4444)×(.3750)×(.4000)}{(.6667)×(.1667)×(.7500)}\approx{\frac{4}{9}×\frac{3}{8}×\frac{2}{5}×\frac{3}{2}×\frac{6}{1}×\frac{4}{3}}=\frac{4×3×2×3×6×4}{9×8×5×2×1×3}\)
Now, don't multiply these numbers out. You are just going to get another enormous fraction. Instead, look for common factors in the top and bottom of the fraction and cancel them. This is a strategy I call "Divide and Conquer" in my classes. Thus,
\(\require{cancel}\frac{4×3×2×3×6×4}{9×8×5×2×1×3}=\frac{\cancel{4}×\cancel{3}×\cancel{2}×\cancel{3}×\cancel{6}×4}{\cancel{9}×\cancel{8}×5×\cancel{2}×\cancel{1}×\cancel{3}}=\frac{4}{5}\)
Using our fractions-to-decimals conversion table, we can convert the \(\frac{4}{5}\) back into its decimal form, \(0.8\). The answer is "B".
Now, let's look back on this problem through the lens of strategy. As you study for the GMAT, you don't want to just look at practice questions from the perspective of "How do I solve this problem?" Instead, use each problem to teach you strategic ways of thinking about entire categories of questions, using the leverage and structure of the problem to train your GMAT reflexes. On this question, we see three primary strategies. One of them is what I call "Mathugliness" -- the way the GMAT deliberately makes questions look harder than they are by giving you ugly math. When you see something like that, don't panic. Party. There has to be an alternative way of looking at the math or simplifying your approach. In this case, there is: convert decimals to fractions ("Fractions Are Your Friends") and then cancel out factors that are common to both numerators and denominators ("Divide and Conquer").
Attachments
File comment: Fraction to Decimal Conversion Table from "Aaron's Cheat Sheets"
Decimal2Fraction.PNG [ 19.51 KiB | Viewed 6511 times ]
General Discussion
26 Dec 2019, 06:11
Kudos
Bookmarks
Hi sjuniv32 ..
I am not sure whether you like my approach. By scanning, I can see 4444, 6667~6666, 7500 is 2*3750 ..
Eq. - \(\frac{(.4444)×(.3750)×(.4000) }{ (.6667)×(.1667)×(.7500)}\)
Dividing/cancelling - 4444 and 6667 by 1111 and 3750 with 7500
\(\frac{4×1 ×(.4000) }{ 6×(.1667)×2}\)
\(\frac{(.4000) }{ 3×(.1667)}\)
Now, we know, that 16.5 * 3 will be approx. equal to 50. The only suitable option is 'B'. You can also write as: \(\frac{(.4000) }{ (.5000)}\)
I am not sure whether you like my approach. By scanning, I can see 4444, 6667~6666, 7500 is 2*3750 ..
Eq. - \(\frac{(.4444)×(.3750)×(.4000) }{ (.6667)×(.1667)×(.7500)}\)
Dividing/cancelling - 4444 and 6667 by 1111 and 3750 with 7500
\(\frac{4×1 ×(.4000) }{ 6×(.1667)×2}\)
\(\frac{(.4000) }{ 3×(.1667)}\)
Now, we know, that 16.5 * 3 will be approx. equal to 50. The only suitable option is 'B'. You can also write as: \(\frac{(.4000) }{ (.5000)}\)
sjuniv32
