Lesson: Percent Increases and Decreases

Comment on Percent Increases and Decreases

maybe another way to think about it

1*1.2*0.85 = 1.02

therefore 2% increase

gmat-admin's picture

Yes, that's basically what we're doing in the video. Except we're starting with 100 rather than 1.

Hi Brent,

Please help me with the solution:

Of the 3,600 employees of Company X, 1/3 are clerical. If the clerical staff were to be reduced by 1/3 what percent of the total number of the remaining employees would then be clerical?
(A) 25 %
(B) 22.2 %
(C) 20 %
(D) 12.5 %
(E) 11.1 %
gmat-admin's picture

If 1/3 of the 3,600 employees are clerical, then there are 1200 clerical workers.

If 1/3 of the clerical staff are laid off, then 400 of the 1200 clerical workers are laid off.

This means there are 800 clerical workers remaining.

Since 400 workers were laid off, the NEW employee population = 3600 - 400 = 3200

"What percent of the total number of the remaining employees would then be clerical?"

Of the 3200 workers, 800 are clerical workers

800/3200 = 1/4 = 25%

Answer: A

The cost C of manufacturing a certain product can be estimated by the formula C = 0.03 rst², where r and s are the amounts, in pounds, of the two major ingredients and t is the production time, in hours. If r is increased by 50 %, s is increased by 20 %, and t is decreased by 30 %, by approximately what percent will the estimated cost of manufacturing the product change ?

(A) 40% increase
(B) 12% increase
(C) 4% increase
(D) 12% decrease
(E) 24% decrease
gmat-admin's picture

Let's plug in some nice values of r, s and t, and then see what happens when we make the given changes.

Let's try r = 10, s = 10 and t = 10

So, cost = 0.03rst² = 0.03(10)(10)(10²) = 3000

If r is increased by 50 %, then r = 15
If s is increased by 20 %, then s = 12
If t is decreased by 30 %, then t = 7
So, NEW cost = 0.03rst² = 0.03(15)(12)(7²)

Let's use some ESTIMATION....

= (0.03)(180)(7²)
= (3/100)(180)(7²)
= (540/100)(7²)
≈ (5.5)(7²)
≈ (5.5)(50)
≈ 275

So, the original cost was $300 and the new cost is $275

This is a DECREASE, so eliminate A, B and C

Percent change = (100)(change)/original
= (100)(300 - 275)/300
= 25/3
≈ 8 1/3%
≈ 8.333%

The closest answer is D, so it must be correct.

Dear Brent,

Thank you for the reinforcement activity questions. They represent a great pool of questions for various difficulty levels. Though I was able to solve (and understand the explanation for) most of the questions above, this particular question perplexed me and I am still unable to understand the explanations provided underneath the question. Kindly help me understand it in a better and simpler manner if possible. Thanks.

Here's the question for your reference:

A certain salesman's yearly income is determined by a base salary plus a commission on the sales he makes during the year. Was the salesman's commission larger than his base salary last year?

(1) If the amount of the commission had been 30 percent higher, the salesman's total income (salary plus commission) would have been 10 percent higher last year.

(2) The absolute difference between the amount of the salesman's base salary and the amount of the commission was equal to 50 percent of the salesman's base salary last year.

Thanks and Regards,
Aashay
gmat-admin's picture

Glad to help.
Here's my step-by-step solution: http://www.beatthegmat.com/income-t285243.html#798714

Cheers,
Brent

Yulia's picture

Hi Brent,

Honestly, I could not rephrase the question the way you did. Could you please explain in a very simple manner how you arrive to "Is C greater than B?". Any advise how to approach such questions.

Target question: Was the salesman's commission larger than his base salary last year?
This is a good candidate for rephrasing the target question.

Let B = base salary last year
Let C = commission last year
So, B+C = TOTAL income last year
REPHRASED target question: Is C greater than B?
gmat-admin's picture

Think I made things confusing when I added the line "So, B+C = TOTAL income last year"

Let B = base salary last year
Let C = commission last year

Target question: Was the salesman's commission larger than his base salary last year?
In other words: Was the commission greater than the base salary?
In other words: Was C greater than B?

Does that help?

Hi Brent, Could you please help me solve this question by assuming some values?

The annual rent collected by a corporation from a certain building was x percent more in 1998 than in 1997 and y percent less in 1999 than in 1998. Was the annual rent collected by the corporation from the building more in 1999 than in 1997?
(1) x > y
(2) xy/100 < x - y
gmat-admin's picture

There's a nice discussion (by several experts) here: http://www.beatthegmat.com/og-13-127-t288372.html

If you have any questions about those approaches, I'm happy to help.

Cheers,
Brent

Sir is thera any other easy way to remember the question

By approximately what percent is x greater than 4/5 if (4/5)(x) = 1?

A. 73%
B. 56%
C. 41%
D. 37%
E. 29%
gmat-admin's picture

Here's one approach:

Given: (4/5)(x) = 1
Divide both sides by 4/5 to get: x = 5/4

So, one option is to determine by what percent 5/4 is greater than 4/5
However, working with fractions can be a pain, so we could also go this route:
5/4 = 25/20 and 4/5 = 16/20

Since both values have the same denominator, we can determine by what percent 25 is greater than 16 (a little easier)
This is the same as the percent increase from 16 to 25

Percent increase = (100)(new - old)/old
= (100)(25 - 16)/16
= (100)(9)/16
≈ 56%

Answer: B

https://gmatclub.com/forum/each-year-for-4-years-a-farmer-increased-the-number-of-trees-in-a-135487.html

Hi Brent,
I find it difficult to solve this problem. I am not clear with the successive percent change calculations. Could you please help me out? Also if you have answered this question elsewhere please share the link.

Thanks in advance.
gmat-admin's picture

Here's my solution: https://gmatclub.com/forum/each-year-for-4-years-a-farmer-increased-the-...

Please let me know if you'd like me to elaborate on any parts of my solution.

Cheers,
Brent

A merchant discounted the sale price of a coat and the sale price of a sweater. Which of the two articles of clothing was discounted by the greater dollar amount?

(1) The percent discount on the coat was 2 percentage points greater than the percent discount on the sweater.
(2) Before the discounts, the sale price of the coat was $10 less than the sale price of the sweater.
gmat-admin's picture

https://gmatclub.com/forum/a-toy-store-regularly-sells-all-stock-at-a-discount-of-103585.html

A toy store regularly sells all stock at a discount of 20 percent to 40 percent. If an additional 25 percent were deducted from the discount price during a special sale, what would be the lowest possible price of a toy costing $16 before any discount?

A. $5.60
B. $7.20
C. $8.80
D. $9.60
E. $15.20

Hi Brent, could you please offer some insight for this question.

Firstly, the initial approach I took to solve this question was that in order to reverse the discounts, you would have to multiply the discounted price by (1+Discount %). I.E. = 16 * (140/100) * (125/100).
This implying that the INITIAL PRICE was reduced by 40%, then 25%. However this approach is wrong and I am not sure why.

The second part of this question which confuses me are the answer choices. The question stem specifically states:

"What would be the lowest possible price of a toy costing $16 before any discount?"

IF the cost is $16 after discount, how could it possibly be any of these answer choices??
A. $5.60
B. $7.20
C. $8.80
D. $9.60
E. $15.20

IF we are trying to find the cost of the Toy "before" discount, this price we find would be higher than $16.

Thank you and apologies if I overlooked anything and made a simple error.

gmat-admin's picture

This is a bit of a Sentence Correction question.

It all comes down to the interpretation of the line "What would be the lowest possible price of a toy costing $16 before any discount?"

Notice that "before any discount" is NEXT TO "costing $16"
This tells us that the before-discount cost is $16
So, the after-discount price will be less than $16

You're reading the line as "Before any discounts, what would be the lowest possible price of a toy costing $16?"
In this case, the answer would be greater than $16

Does that help?

Here's my solution: https://gmatclub.com/forum/a-toy-store-regularly-sells-all-stock-at-a-di...

Cheers,
Brent

Hi Brent, could you please help to provide solution to this DS question:

Jason’s salary and Karen’s salary were each p percent greater in 1998 than in 1995. What is the value of p ?

(1) In 1995 Karen’s salary was $2,000 greater than Jason’s.
(2) In 1998 Karen’s salary was $2,440 greater than Jason’s.

Thanks!
gmat-admin's picture

You bet!
Here's my full solution: https://gmatclub.com/forum/janson-s-salary-and-karen-s-salary-were-each-...

Cheers,
Brent

Company P had 15 percent more employees in December than it had in January. If Company P had 460 employees in December, how many employees did it have in January?

(A) 391
(B) 400
(C) 410
(D) 423
(E) 445

Thank you Brent for explaining the common mistake. I would like to know how did you get to 1.15.... please explain the rule because I want to understand it

Thank you for your prompt help
Fatima-Zahra
gmat-admin's picture

My solution: https://gmatclub.com/forum/company-p-had-15-percent-more-employees-in-de...

If we let J = # of employees in January, then...

# of employees in Dec = (number employees in Jan) + (15% of the number employees in Jan)
= (J) + (15% of J)
= J + 0.15J
= 1.15J

Let's examine one more example.

y is 37% greater than x
We can write: y = x + (37% of x)
y = x + 0.37x
y = 1.37x

Does that help?

Cheers,
Brent

Martha's Hair Saloon has recently lowered the prices of haircuts. If the decrease in the price of a haircut is 20% of the new price of a haircut, what is the approximate percent decrease in the price of a haircut at Martha's ?

A. 15%
B. 16.7%
C. 20%
D. 25%
E. 87.5%

Hi Brent, could you help me out in this question with the input/ output method please? Thank you before hand!
gmat-admin's picture

The INPUT-OUTPUT method is used when we have a Variables in the Answer Choices question (aka VIAC).
So, that approach wouldn't work here.

For more on VIAC's, watch: https://www.gmatprepnow.com/module/gmat-word-problems/video/933

Here's my full solution: https://gmatclub.com/forum/martha-s-hair-saloon-has-recently-lowered-the...

Cheers,
Brent

Hi Brent,

question link https://gmatclub.com/forum/the-output-of-a-factory-was-increased-by-10-to-keep-up-with-138775.html

So I am a little confused with formula % Change=(new-original)/original. Am I understanding it correct that this formula works when we are trying to calculate percent INCREASE. When we are going the opposite direction and are trying to calculate the percent decrease from a number than instead of dividing difference by original number, we must divide difference by new number (in case of this question we are trying to calculate by how much percent we have to decrease a number (say original number is 100 and final number is 132), it is the difference in relation to 132, not 100. So we are dividing 32/132. Am I getting the concept correct? Otherwise, I do not understand why we are dividing 32 by 132, not the 100.

Thanks a bunch!
gmat-admin's picture

Question link: https://gmatclub.com/forum/the-output-of-a-factory-was-increased-by-10-t...

In general, percent increase or percent decrease = (100)(change)/(old value)

This question is somewhat confusing, since we already increased the output twice, and now we're trying to get back to the original value.

However, our goal is to go from 132 to 100.
So, 132 is the OLD value, and 100 is the NEW value.

So, percent decrease = 100(132-100)/132

Does that help?

Cheers,
Brent

Hi Brent,

Could you please help with the following question?

The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A present and inversely proportional to the concentration of chemical B present. If the concentration of chemical B is increased by 100 percent, which of the following is closest to the percent change in the concentration of chemical A required to keep the reaction rate unchanged?

100% decrease

50% decrease

40% decrease

40% increase

50% increase

Thanks,
Kevin
gmat-admin's picture

Tough question!!!

From the given information, we can write: Rate of reaction = A²/B

Let's plug in some INITIAL values.
Let's say the concentration of chemical A present = 6
Let's say the concentration of chemical B present = 2
With these values, the rate of reaction = 6²/2 = 36/2 = 18

Now let's see what happens when the concentration of chemical B is increased by 100 percent.
This means the concentration of chemical B increases to 4
So, the rate of reaction = A²/4
We want the rate to REMAIN at 18
So, we get the equation A²/4 = 18

Solve for A by first multiplying both sides by 4 to get: A² = 72
Solve: A = √72
So, in order for the rate to stay at 18, A must equal √72

We know that √64 = 8 and √81 = 9
Since 72 is roughly halfway between 64 and 81, we can conclude that √72 is roughly halfway between 8 and 9
So, let's say A = 8.5

So, A must increase from 6 to 8.5

Percent increase = 100(8.5 - 6)/6
= 100(2.5)/6
= 250/6
≈ 40%

Answer: D

Cheers,
Brent

Hi Brent,

Thanks for the fast and detailed solution!

What got me confused was the term "concentration" not referring to a percentage.
Is it used for percentages as well as any other number regularly in the GMAT or is this question an exception?

Cheers,
Kevin
gmat-admin's picture

Good question, Kevin.

Concentration can be measured in many different ways (e.g., by percent, parts per million, grams per liter, etc).
On some occasions, the GMAT test-makers will provide units of measurement, and on other occasions, they don't.

For example, with Geometry questions, the test-makers will often drop the units of measurement and just say "some line has length 10" or "rectangle R has area 6", etc. Other times, they'll include units, as in "the flask contains 3 gallons of water."

Cheers,
Brent

Thank you for the explanation and examples!

official guide question 650-800 range, link is not available. kindly update this.

https://gmatclub.com/forum/janson-s-salary-and-karen-s-salary-were-each-p-percent-104127-20.html

gmat-admin's picture

https://gmatclub.com/forum/by-approximately-what-percent-is-x-greater-than-4-5-if-4-5-x-122307.html

Notice that 1.2 is 50% greater than 0.8.....

How can you get percent? Would you please explain that?
gmat-admin's picture

Link: https://gmatclub.com/forum/by-approximately-what-percent-is-x-greater-th...

50% of 0.8 = 0.4
Since 1.2 = 0.8 + 0.4, we can say that 1.2 = 0.8 + (50% of 0.8)
In other words, 1.2 is 50% GREATER THAN 0.8

Does that help?

Cheers,
Brent

Question link: https://gmatclub.com/forum/a-certain-salesman-s-yearly-income-is-determined-by-a-base-salary-plus-126533.html

Hi Brent,
In statement 2 of this question, how do we know that the statement is talking about "absolute" value?
gmat-admin's picture

Question link: https://gmatclub.com/forum/a-certain-salesman-s-yearly-income-is-determi...

(2) The difference between the amount of the salesman's base salary and the amount of the commission was equal to 50 percent of the salesman's base salary last year.

We know that the salesman's base salary last year must be positive.
So, (salesman's base salary) - (commission) must be positive
To ensure that this amount is positive, we can take the absolute value of it.

Does that help?

Cheers,
Brent

Yulia's picture

Hi Brent,

How did you see that "Did the salesman's base salary account for more than half of the salesman's yearly income last year?" = C greater than B?
I saw it this way. Yearly income (Y) = base salary (B) + commission (C) -> target question B= 1/2 x Y
Where am i making a mistake?
A certain salesman's yearly income is determined by a base salary plus a commission on the sales he makes during the year. Did the salesman's base salary account for more than half of the salesman's yearly income last year?

(1) If the amount of the commission had been 30 percent higher, the salesman's income would have been 10 percent higher last year.

(2) The difference between the amount of the salesman's base salary and the amount of the commission was equal to 50 percent of the salesman's base salary last year.
gmat-admin's picture

Let B = base salary last year
Let C = commission last year

We know that the salesman's total income = B + C

Let's see what would happen IF exactly HALF of his total income was from his BASE salary.
This would mean that the other HALF of his income was from COMMISSIONS
In this scenario, B = C

Now let's see what would happen IF 1/3 of his total income was from his BASE salary.
This would mean that the other 2/3 of his income was from COMMISSIONS
In this scenario, B < C

Now let's see what would happen IF 9/10 of his total income was from his BASE salary.
This would mean that the other 1/2 of his income was from COMMISSIONS
In this scenario, B > C

So, if his BASE salary his MORE THAN half of his total income, it must be the case that B > C

NOTE: This is the point in my solution when I realized that I SHOULD have written (Is B greater than C) in my original solution.
I've now edited that part of my solution (although the outcome is still the same)

Does that help?

Hi Brent,

I actually understand this question’s concept just fine. I got stuck with timing on 460/1.15. The long division pushed me over the time limit. What is your quickest shortcut to avoid the long division here? I think it’ll help me apply it to other problems.

Company P had 15 percent more employees in December than it had in January. If Company P had 460 employees in December, how many employees did it have in January?
gmat-admin's picture

Question link: https://gmatclub.com/forum/company-p-had-15-percent-more-employees-in-de...

Great question!
Here are two possible approaches (other than long division) we could take when evaluating 460/1.15

1) Create equivalent fractions by multiply numerator and denominator by the same value.
For example we could take 460/1.15 and multiply top and bottom by 2 to get: 920/2.3
If we're lucky, we might notice that 2.3 (from this new fraction) divides nicely into 460 (from our original fraction)
If we noticed this then we can rewrite 920 to get:
920/2.3 = (2 x 460)/2.3
= (2/1)(460/2.3)
= (2)(200)
= 400

2) Rewrite 1.15 as a fraction
1.15 = 115/100 = 23/20
So, 460/1.15 = 460/(23/20) = (460)(20/23) = (20)(20/1) = 400
Aside: The GMAT often rewards students (time-wise) for using fractions.

I hope that helps.

Cheers,
Brent

Hi Brent,
Pls explain me this problem
In a factory that produces computer circuit boards, 4.5 percent of all boards produced are found to be defective and are repaired before being sold, but 10 percent of all defective boards are sold without being repaired. What percentage of boards produced in the factory are defective?

A. 4.5%
B. 5.0%
C. 6.0%
D. 10.0%
E. 14.5%
gmat-admin's picture

It turns out we can use the Double Matrix method to solve this question!
Here's my full solution: https://gmatclub.com/forum/in-a-factory-that-produces-computer-circuit-b...

Cheers,
Brent

Hi Brent,

What about questions related to markup pricing? do we get such questions on the GMAT? if so, how do we solve such questions?
gmat-admin's picture

You'll find many such questions in the linked questions above (in the Reinforcement Activities box).

For additional questions, try this link: https://gmatclub.com/forum/problem-solving-ps-140/?start=0&selected_tags...
The link filters out all GMAT Club questions involving percent. Some of those questions are related to markup (and markdown) pricing.

I hope that helps.

For the first question in the video, I don't think that 100 matters,

I just use 12/40 -> 6/20, 6*5 = 30
gmat-admin's picture

There are many different ways to convert 12/40 to a percent.

In a way, you're solution still uses 100.
That is, you're taking 6/20 and multiplying numerator and denominator by 5 to get 30/100, which equals 30%

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