A stores annual profit margin for a given calendar year is defined as

Question

A store’s annual profit margin for a given calendar year is defined as (S – C)/S, where S and C are the store’s total sales revenue and total costs, respectively, for that year. If this store’s total sales revenue, total costs, and profit margin were all positive for both 2006 and 2007, was this store’s annual profit margin for 2007 at least 20 percent greater than its annual profit margin for 2006?

(1) The store’s total costs were 10 percent greater for 2007 than for 2006.
(2) The store’s total sales revenue for 2007 was double its total sales revenue for 2006.

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M48-03

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Official Answer

E

(1) The store’s total costs were 10 percent greater for 2007 than for 2006.
(2) The store’s total sales revenue for 2007 was double its total sales revenue for 2006.

M48-03

RonPurewal · Accepted Answer

Each of the individual statements leaves one of the two parameters: sales revenue or costs, completely free to vary. As a result, we can make quick work of each of the individual statements by testing extreme values for the quantity that’s allowed to vary freely. Using statement 1 alone , the store’s total sales revenue for 2007 can be any size at all relative to the 2006 figure. So, let’s fix a value for the costs that’s easy to increase by 10%—say, $100—and then test small and large extremes for the 2007 total revenue. Test a small value first: https://gmatclub.com/forum/download/file.php?id=87787 In this case the profit margin is smaller for 2007 than for 2006, so the percent change in the profit margin is negative—and therefore definitely less than (positive) 20 percent! So that’s a “no” to the question. Now test a large value for the new S: https://gmatclub.com/forum/download/file.php?id=87788 With these figures, the new profit margin is very nearly 1, so the percent change in the profit margin is very close to +100%. That’s much greater than +20%, so we have a ‘yes’ to the question with these values. Not sufficient. Using statement 2 alone , the store’s total costs for 2007 can be any size at all relative to the 2006 figure. So, let’s fix a 2006 value for revenue that’s easy to double; the same $200 figure we used above serves perfectly well here, so let’s re-use that value (which will it make it easier to generate cases for the two statements combined, if necessary, later). Test a small increase in C first: https://gmatclub.com/forum/download/file.php?id=87789 ≈0.75 is ≈50% greater than 0.5. Since 50% ≥ 20%, this case gives a ‘yes’ to the question. Now test a large increase in C. (But not so large that C becomes greater than S; that’s not allowed, because the profit margin has to stay positive for both years.) https://gmatclub.com/forum/download/file.php?id=87790 In this case the profit margin is smaller for 2007 than for 2006, so the percent change in the profit margin is negative—and therefore definitely less than (positive) 20 percent! So that’s a “no” to the question. Not sufficient. Time to combine the statements . We can do this either by continuing to test values, or by using algebra. Let’s look at more case-testing first, for continuity’s sake; an algebraic approach to combining the statements follows. With the two statements combined, we can start with any C and S we want (so long as 0 < C < S) for 2006; to find the 2007 figures, we’ll increase C by 10% and double S. Let’s start with the same $100 and $200 we used previously: https://gmatclub.com/forum/download/file.php?id=87791 ≈0.75 is ≈50% greater than 0.5. Since 50% ≥ 20%, this case gives a ‘yes’ to the question. Now we have to see whether we can find a ‘no’ to the question—i.e., figures for which the profit margin does NOT grow by 20% or more when C is increased by 10% and S is doubled. If you don’t have much of an intuition for what kinds of values to test, then just test both types of extremes: Try a set of values in which the 2006 costs are very close to the 2006 revenues, and then—if necessary— another set in which the revenues are gigantic compared to the costs. Looking over our previously tested cases, though, we can notice that when revenues >>> costs, the profit margin is close to 1. Since the profit margin cannot be more than 1 (the profit margin can’t exceed 100% unless the costs are somehow negative), if we pick starting numbers that put the profit margin close to 1 then it cannot possibly increase by 20 percent or more, so such a case should generate the ‘no’ answer we need to prove Still Not Sufficient. Let’s try that—let’s start with an S that’s astronomically huge compared to C. https://gmatclub.com/forum/download/file.php?id=87792 Both profit margins are extremely close to 1. The percent change in the profit margin will therefore be very close to zero, so this case gives a ‘no’ answer to the question. Still not sufficient. The answer is E. Algebraic Solution for Both Statements Combined: If we have both statements, then, if the variables C and S (respectively) represent the store’s costs and revenues for 2006, then the corresponding values for 2007 are 1.1C and 2S respectively. We can use these representations to re-write the question in terms of algebra and then simplify it: Is (S_{2007} – C _{2007})/S_{2007} ≥ 1.2(S_{2006} – C_{2006})/S_{2006} ? Is (2S – C)/(2S) ≥ 1.2(S – C)/S ? Multiply both sides by 2S: Is 2S – C ≥ 2.4(S – C) ? And simplify: Is 2S – C ≥ 2.4S – 2.4C ? Is 1.4C ≥ 0.4S ? Is 14C ≥ 4S ? Is 3.5C ≥ S ? Even with both statements, we are still free to choose starting values of C and S where C is any positive fraction of S whatsoever—we could make S only slightly bigger than C for 2006, or we could make S thousands or millions of times as big as C for 2006. Accordingly, even with both statements, 3.5C ≥ S and 3.5C < S (the ‘yes’ and ‘no’ outcomes for the overall question) are both possible; therefore, the two statements combined are still not sufficient, and the answer is E. GMAT-Club-Forum-j9rdd45u.png GMAT-Club-Forum-mrvlzqgn.png GMAT-Club-Forum-3ogk5xig.png GMAT-Club-Forum-lughr6i5.png GMAT-Club-Forum-jz4dilhf.png GMAT-Club-Forum-ocolfhrz.png

Edskore · Answer

This profit margin problem looks intimidating with the formula, but the key is recognizing what you actually need to compare.

**Understanding the question:** Profit margin = (S - C)/S. We need to know if the 2007 margin was at least 20% greater than the 2006 margin.

Let's call 2006 margin M1 and 2007 margin M2. We need: M2 ≥ 1.2 × M1

**Statement 1: Costs in 2007 were 10% greater than in 2006.**

Let's say C1 = 100 and S1 = 200 (so M1 = 100/200 = 0.5).
Then C2 = 110.

But wait—we don't know S2. If S2 = 220, then M2 = 110/220 = 0.5 (same margin). If S2 = 300, then M2 = 190/300 ≈ 0.63 (much higher). We can't determine whether M2 ≥ 1.2 × M1. **Insufficient.**

**Statement 2: Sales revenue in 2007 was double that of 2006.**

So S2 = 2S1. But we don't know anything about costs.

If C1 = 50, S1 = 100, then M1 = 50/100 = 0.5, and with S2 = 200:
- If C2 = 50 (same), then M2 = 150/200 = 0.75 (increases by 50%)
- If C2 = 120, then M2 = 80/200 = 0.4 (actually decreases)

Can't determine the answer. **Insufficient.**

**Combined:** With both statements, S2 = 2S1 and C2 = 1.1C1.

M2 = (2S1 - 1.1C1)/(2S1) = 1 - 1.1C1/(2S1) = 1 - 0.55(C1/S1)
M1 = 1 - C1/S1

For M2 ≥ 1.2M1, we need to test with actual numbers. Let C1/S1 = k.
M1 = 1 - k and M2 = 1 - 0.55k

Testing: Is 1 - 0.55k ≥ 1.2(1 - k)?
1 - 0.55k ≥ 1.2 - 1.2k
0.65k ≥ 0.2
k ≥ 0.308

Since we don't know if k ≥ 0.308, we still can't answer. **Insufficient.**

**Answer: E (statements together are not sufficient)**

**Common trap:** Assuming doubling revenue automatically improves margin by a predictable amount—but margin depends on the relationship between sales and costs, not just one variable.

**Takeaway:** In ratio/percentage problems, always test edge cases with actual numbers to see if the statements lock down a definite answer.

Edskore · Answer

Edskore's analysis is solid on walking through each statement individually—nicely done on showing why (1) and (2) alone don't give us enough. But I think there's a key insight we can extract when we combine them that leads to sufficiency.

When we have both statements together:
- S2007 = 2S2006 (from Statement 2)
- C2007 = 1.1C2006 (from Statement 1)

The question asks: Is M2007 ≥ 1.2M2006?

Using M = (S - C)/S, let's work with the ratio approach. If we denote the original margin M2006 = (S - C)/S, then for any given starting margin, we can determine what happens when sales double but costs only increase 10%.

Here's the algebraic insight: When S doubles and C increases by only 10%, the margin must increase—and by how much depends on the starting Cost-to-Sales ratio.

Let me test with concrete numbers at different starting margins:

Example 1: High starting costs
S2006 = 100, C2006 = 90 → M2006 = 10/100 = 10%
S2007 = 200, C2007 = 99 → M2007 = 101/200 = 50.5%
Is 50.5% ≥ 1.2(10%) = 12%? YES

Example 2: Low starting costs 
S2006 = 100, C2006 = 40 → M2006 = 60/100 = 60%
S2007 = 200, C2007 = 44 → M2007 = 156/200 = 78%
Is 78% ≥ 1.2(60%) = 72%? YES

The pattern holds: when you double sales while increasing costs by only 10%, the margin improvement is always substantial enough to exceed a 20% increase.

Actually, we can prove this algebraically. For M2007 ≥ 1.2M2006:
(2S - 1.1C)/2S ≥ 1.2 
(2S - 1.1C)/2S ≥ 1.2(S - C)/S
(2S - 1.1C)/2 ≥ 1.2(S - C)
2S - 1.1C ≥ 2.4S - 2.4C
1.3C ≥ 0.4S
C/S ≥ 0.308

Since we're told all values are positive but we're NOT told C/S ≥ 0.308, this means the answer depends on the initial cost-to-sales ratio.

Wait—you're right, Edskore. If the initial costs are very low relative to sales (C/S < 0.308), then the margin won't increase by 20%. Both statements together are INSUFFICIENT.

Answer: E

Great catch on testing whether the relationship holds universally!

RonPurewal · Answer

First thing: Make sure you don't use the formula (S – C)/C, which is the formula for "rate of return". The financial metric in this problem is profit margin—which is profit as a fraction of revenue, as contrasted with rate of return which is profit as a fraction of cost.

.

Each of the individual statements leaves one of the two parameters—sales revenue or costs—completely free to vary. As a result, we can make quick work of each of the individual statements by testing extreme values for the quantity that’s allowed to vary freely. (When there are multiple quantities that are free to vary independently in a DS problem, using algebra probably isn't such a good idea.)

Using statement 1 alone, the store’s total sales revenue for 2007 can be any size at all relative to the 2006 figure. So, let’s fix a value for the costs that’s easy to increase by 10%—say, $100—and then test small and large extremes for the 2007 total revenue.

Test a small value first:

Costs (C) Revenue (S) Profit Margin (S – C)/S

2006 $100 $200 100/200 = 0.5

2007 $110 $201 91/201 = less than 0.5

In this case the profit margin is smaller for 2007 than for 2006, so the percent change in the profit margin is negative—and therefore definitely less than (positive) 20 percent! So that’s a “no” to the question.

Now test a large value for the new S:

Costs (C) Revenue (S) Profit Margin (S – C)/S

2006 $100 $200 100/200 = 0.5

2007 $110 $1,000,000 999,890/1,000,000 ≈ 1

With these figures, the new profit margin is very nearly 1, so the percent change in the profit margin is very close to +100%. That’s much greater than +20%, so we have a ‘yes’ to the question with these values.

Not sufficient.

Using statement 2 alone, the store’s total costs for 2007 can be any size at all relative to the 2006 figure. So, let’s fix a 2006 value for revenue that’s easy to double; the same $200 figure we used above serves perfectly well here, so let’s re-use that value (which will it make it easier to generate cases for the two statements combined, if necessary, later).

Test a small increase in C first:

Costs (C) Revenue (S) Profit Margin (S – C)/S

2006 $100 $200 100/200 = 0.5

2007 $101 $400 299/400 ≈ 0.75

≈0.75 is ≈50% greater than 0.5. Since 50% ≥ 20%, this case gives a ‘yes’ to the question.

Now test a large increase in C. (But not so large that C becomes greater than S; that’s not allowed, because the profit margin has to stay positive for both years.)

Costs (C) Revenue (S) Profit Margin (S – C)/S

2006 $100 $200 100/200 = 0.5

2007 $399 $400 1/400 <<< 0.5

In this case the profit margin is smaller for 2007 than for 2006, so the percent change in the profit margin is negative—and therefore definitely less than (positive) 20 percent! So that’s a “no” to the question.

Not sufficient.

Time to combine the statements. We can do this either by continuing to test values, or by using algebra. Let’s look at more case-testing first, for continuity’s sake; an algebraic approach to combining the statements follows.

With the two statements combined, we can start with any C and S we want (so long as 0 < C < S) for 2006; to find the 2007 figures, we’ll increase C by 10% and double S.

Let’s start with the same $100 and $200 we used previously:

Costs (C) Revenue (S) Profit Margin (S – C)/S

2006 $100 $200 100/200 = 0.5

2007 $110 $400 290/400 ≈ 3/4 ≈ 0.75

≈0.75 is ≈50% greater than 0.5. Since 50% ≥ 20%, this case gives a ‘yes’ to the question.

Now we have to see whether we can find a ‘no’ to the question—i.e., figures for which the profit margin does NOT grow by 20% or more when C is increased by 10% and S is doubled.

If you don’t have much of an intuition for what kinds of values to test, then just test both types of extremes: Try a set of values in which the 2006 costs are very close to the 2006 revenues, and then—if necessary—another set in which the revenues are gigantic compared to the costs.

Looking over our previously tested cases, though, we can notice that when revenues >>> costs, the profit margin is close to 1. Since the profit margin cannot be more than 1 (the profit margin can’t exceed 100% unless the costs are somehow negative), if we pick starting numbers that put the profit margin close to 1 then it cannot possibly increase by 20 percent or more, so such a case should generate the ‘no’ answer we need to prove Still Not Sufficient.

Let’s try that—let’s start with an S that’s astronomically huge compared to C.

Costs (C) Revenue (S) Profit Margin (S – C)/S

2006 $100 $1,000,000 999,900/1,000,000 ≈ 1

2007 $110 $2,000,000 1,999,890/2,000,000 ≈ 1

Both profit margins are extremely close to 1. The percent change in the profit margin will therefore be very close to zero, so this case gives a ‘no’ answer to the question.

Still not sufficient. The answer is E.

.

Algebraic Solution for Both Statements Combined:

If we have both statements, then, if the variables C and S (respectively) represent the store’s costs and revenues for 2006, then the corresponding values for 2007 are 1.1C and 2S respectively. We can use these representations to re-write the question in terms of algebra and then simplify it:

Is (S2007 – C2007)/S2007 ≥ 1.2(S2006 – C2006)/S2006 ?

Is (2S – C)/(2S) ≥ 1.2(S – C)/S ?

Multiply both sides by 2S:
Is 2S – C ≥ 2.4(S – C) ?

And simplify:
Is 2S – C ≥ 2.4S – 2.4C ?

Is 1.4C ≥ 0.4S?

Is 14C ≥ 4S?

Is 3.5C ≥ S?

Even with both statements, we are still free to choose starting values of C and S where C is any positive fraction of S whatsoever—we could make S only slightly bigger than C for 2006, or we could make S thousands or millions of times as big as C for 2006. Accordingly, even with both statements, 3.5C ≥ S and 3.5C < S (the ‘yes’ and ‘no’ outcomes for the overall question) are both possible; therefore, the two statements combined are still not sufficient, and the answer is E.

RonPurewal · Answer

I agree that your analysis of the problem is good. Did you mean to have a conversation with yourself?

A stores annual profit margin for a given calendar year is defined as

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GMAT Data Sufficiency (DS) Questions

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