Lesson: Percent Increases and Decreases

Comment on Percent Increases and Decreases

At minute 9:16, why is 30% 0.7 and 0.3? what am I missing?
greenlight-admin's picture

There are 2 ways to handle a decrease of 30%. We can find 30% of the original value and then subtract that amount from the original value, or we can just find 70% of the original value.

Here's another way to think of it. Let's say there are 100 people in a room. If 30% of those people leave, how many are remaining? Well, if 30% have left, then we know that 70% of the people are remaining. 70% of 100 = 70 people are remaining. Alternatively, 30% of 100 = 30. 100 - 30 = 70. Same answer, two ways to think of it.

I am having a hard time understanding when to apply the different formulas (mainly (1+/- %change/100)*100). I tried solving the wolf population problem by using proportions, but it didn't work -

x = 300%
60 100%

x=180

Can you tell me why this doesn't work, and how to know when to use the formula?
greenlight-admin's picture

I believe you are confusing "300% of" with "300% increase" and "300% more than"

300% of 50 is 150.
300% of 60 is 180.
300% of 11 is 33.

If 50 is increased by 300%, then we take 50 and add 300% of 50 to it. So, 50 increased by 300% is 200.
Likewise, 60 increased by 300% is 240.
And 11 increased by 300% is 44

On a similar note, 200 is 300% greater than 50.
And 44 is 300% greater than 11
And 4000 is 300% greater than 1000

So, in your approach, you are not treating the increase as a 300% increase. The question tells us that the new population is 300% MORE THAN the old population. However, in your approach, you are saying that the new population is 300% OF the old population. In other words, you are saying that the new population is THREE TIMES old population.

For the question with Hilda and her hats - I am confused with how you decoded the "net result" to mean the change in the wholesale price from the sale price. How come you didn't calculate for the net price from the wholesale price and retail price?
greenlight-admin's picture

You're referring to the question that starts at 3:45 in the video.

The "net result" in this instance refers to the TOTAL (combined) effect of all the changes put together.

For example, if I gave you 8 apples, then took 3 apples away from you, then gave you 2 apples and then took away 1 apple, then the NET RESULT would be the same as giving you 6 apples.

In the Hilda question, increasing the wholesale price by 20% and then decreasing the resulting price by 15% has the same effect (NET RESULT) as increasing the wholesale price by 2%

Does that help?

Dear Brent,

regarding this problem :

http://greprepclub.com/forum/jack-and-jill-each-bought-the-same-tv-set-using-a-10-off-co-3534.html

can you please help me understand why adding 8.5% tax is the same as multiplying by 1.085?

and regarding this one:

http://greprepclub.com/forum/qotd-21-from-2011-to-2012-jack-s-annual-salary-increased-2641.html

I think I missing here the reason why the result is expressed as that Arnie’s salary is 15.8% more than Jack’s salary, rather than that Arnie’ salary is 115.8% of Jack’s salary (what I have got). I have the feeling this may have something to do with the fact that the problem asks me “what percent greater” is Arnie’s salary compared to Jack’s salary. Am I forgetting a fundamental step? Finally, a last doubt, when a question asks me to round to the nearest 0.1%, does it simply mean that I am asked to white a percentage value rounded to the first decimal digit?

Thank you very much. I am sorry that I am writing you so many times.
greenlight-admin's picture

Link to the first question: http://greprepclub.com/forum/jack-and-jill-each-bought-the-same-tv-set-u...

Why is adding 8.5% tax is the same as multiplying by 1.085?

Let x = the PRE-TAX price of the TV
So, 0.08x = the 8% tax on the TV
So, the TOTAL price = x + 0.08x = 1.08x

----------------------------------
Link to second question: http://greprepclub.com/forum/qotd-21-from-2011-to-2012-jack-s-annual-sal...

You're correct when you say that Arnie’ salary IS 115.8% of Jack’s salary. HOWEVER, the question asks "Arnie’s salary was what percent GREATER THAN Jack’s salary?"

Other examples: "X is 116% OF Y" is the same as saying "X is 16% greater than Y"

And "X is 180% OF Y" is the same as saying "X is 80% greater than Y"

And "X is 245% OF Y" is the same as saying "X is 145% greater than Y"
------------------------------

"Finally, a last doubt, when a question asks me to round to the nearest 0.1%, does it simply mean that I am asked to white a percentage value rounded to the first decimal digit?"

That's correct!

Cheers,
Brent

Hi Brent please I have a question in relation to the question above. I did use figures, 100 for both Jack's and Arnie's current salary. Had a difference of 14.4 from their previous salary but did not arrive at the answer when I divided the difference by Arnie's previous salary which was 105.3. Please can you help me understand why?
greenlight-admin's picture

You've done everything correctly so far.
Using $100 for both Jack's and Arnie's 2012 salary, we get the following:

Arnie's 2011 salary = $105.263
Jack's 2011 salary = $90.909

So, we can see that Arnie's salary is $14.354 greater than Jack's salary.

The question asks "Arnie’s annual salary in 2011 was what percent greater than JACK'S annual salary in 2011?

We now know that Arnie’s 2011 salary (of $105.263) is $14.354 greater than JACK'S 2011 salary (of $90.909).

Since we're comparing the salary difference with JACK'S 2011 salary, we must take the fraction $14.354/$90.909 and convert it to a PERCENT.

We get: $14.354/$90.909 = 15.8%

Does that help?

Cheers,
Brent

Yes it does.... thanks I guess it was basically with the wording because I was the percentage with the difference over Arnie’s previous salary

A full glass of juice is a mixture of 20% grape juice and 80% apple juice. The contents of the glass are poured into a pitcher that is 200 percent larger than the glass. The remainder of the pitcher is filled with 16 ounces of water. What was the original volume of grape juice in the mixture?
(A) 1.6 ounce (B) 3.2 ounce (C) 4.8 ounce (D) 6.4 ounce (E) 8 ounces

I solved this question as below-

Given mixture : 20% Grape and 80% Apple juice
Poured in pitcher which is 200% greater than original.
Let the original volume of the mixture be x.
Now: new = (1 + 200/100)*x
new = (1+2)x
we know new which is pitcher, there was 16 ounces water added to fill the remainder of pitcher. So it can be written as original mixture x + 16. Substituting this in above equation we get-
x+16 = (1+2)x ; x+16 = 3x
16=2x x=8
So the original volume of the mixture is 8 ounces. Volume of grape juice is 20% of 8 which is 1.6 ounces. Is this method of solving correct?
greenlight-admin's picture

Hi Deepak,

Your method is perfect - nice work!

Here's my full solution (it's a little different from yours): https://greprepclub.com/forum/a-full-glass-of-juice-is-a-mixture-of-20-g...

Cheers,
Brent

Thank you Brent, I did see it, that is why I asked if mine was correct. :) Thank you so much.

I have a doubt in this question-
From 2011 to 2012, Jack’s annual salary increased by 10 percent and Arnie’s annual salary decreased by 5 percent. If their annual salaries were equal in 2012, then Arnie’s annual salary in 2011 was what percent greater than Jack’s annual salary in 2011?

I did reach till this point where the fraction of Arnie to Jack's salary is : A/J = 1.158. Now Arnie's salary is 1.15 times Jack's salary. So it is 115.8% of Jack's salary. Which means it is 15.8% more than Jack's salary ( if we consider Jack's salary as 100) Is this understanding correct? Because I had a confusion that answer should be 115.8%.

greenlight-admin's picture

That's a perfectly valid approach, Deepak.

Cheers,
Brent

Four feet are cut from a 12-foot board. What is the percent decrease in length?

In this question are feet and foot both same unit? If yes then answer has to be 66.66% whereas the solution is as follows-
12-8/12 *100 = 33.33%
Please suggest.
greenlight-admin's picture

"feet" is the plural of "foot"
1 foot
2 feet
3 feet
However, when used in hyphen form (as in 12-foot board),we use "foot" for everything (yes, English is a lawless language!)

So, if 4 feet are removed from the 12-foot board, the board now has length 8.

Percent decrease = 100(difference in values)/ORIGINAL value

The length changes from 12 to 8 (so the original length
= 100(12 - 8)/12
= 33.333...%

Cheers,
Brent

Thank you Brent :)

Hello Brent,
I get a small confusion in percent change problems.When we say that the price of a pen increased by 20% ,We solve it as taking 20% of pencil's original price.(eg. original price =100$). So 20/100 *100$ = 20$. So the price of pencil increased by 20$ and new price is 100 + 20 = 120$. I understand this.

Now when in other problems like eg, Bob is 30 pounds heavier than Sara. If Bob and Sara each gain 30 pounds, then Bob's weight is 25% greater than Sara's weight. How much does Bob weigh right now?

in this question, I solved it as:
Let Bob's weight = B and sara's = S
B = S+30

Adding 30 to each of their weight:
B+30 = s+60
B+30 = 25/100 + (s+60)

This was incorrect at two places:
1) adding 30 more to sara's weight and making it s+60. Can you explain me why I'm wrong?
2) adding the percentage to Sara's weight. I added 25% to Sara's weight because I understood it as Bob is 25% greater than Sara, so adding 25% to sara's weight would be equal to Bob's weight. I added because, I get confused with the term increase and greater than. When we use the word increases,(price of a pen increases by 20%), we take 20% of pen and add the value to original value. So in the same way, should I take 25% of Sara's weight and add that value to sara's weight to get Bob's weight?

Because the expression in the solution was like:
B+30 = 1.25(S+30)
Please explain. I'm totally confused.
greenlight-admin's picture

If k is 25% greater than x, then we can say: k = 1.25x

Likewise, if k is 45% greater than x, then we can say: k = 1.45x

And, if k is 113% greater than x, then we can say: k = 2.13x

If k is 40% LESS than x, then we can say: k = 0.6x

Now let's examine the question...

Let B = Bob's PRESENT weight
Let S = Sara's PRESENT weight
"Bob is 30 pounds heavier than Sara."
We can write: B = S + 30 (as you have)

Adding 30 to each weight, we get:
Bob's NEW weight = B + 30
Sara's NEW weight = S + 30

ASIDE: If S = Sara's PRESENT weight, then S+30 = her NEW weight.
You said S+60 = Sara's weight, but this represents a weight gain of 60 pounds.

"Bob's weight is 25% greater than Sara's weight."
In other words: (Bob's NEW weight) = 1.25(Sarah's NEW weight)
We can write: B + 30 = 1.25(S + 30)

We now have a system of two equations:
B = S + 30
B + 30 = 1.25(S + 30)

When we solve this system, we get: S = 90 and B = 120

So, Sarah's PRESENT weight is 90 pounds, and Bob's PRESENT weight is 120 pounds

Does that help?

Cheers,
Brent

I understand this. Thank you. So in general it means that, if there is 20% increase in the price of a commodity then the new price is 120% of the old price. The same is case with 20% greater as well. Is this understanding correct?
greenlight-admin's picture

That's correct.

Hi I was doing some practice question here, https://gre.graduateshotline.com/quantitative_reasoning_test.html
but they don't give answers. Could you explain number 9? I get that #8 doesn't give any info on 2010 so we can't answer that but question #9 confused me.

Thanks.
greenlight-admin's picture

Let's say the sales for 2008 totaled $100
So, the sales for 2009 = $105 (after a 5% increase)
And the sales for 2010 = $99.75 (after a 5% decrease)
---------------------------
Question: In store D, the dollar amount of sales for 2009 was approximately what percent of the dollar amount of sales for 2010?
A) 105%
B) 106%
C) 95%
D) 104%
E) It cannot be determined from the information given.
---------------------------
We get: 105/99.75 ≈ 105/100 ≈ 105%

Answer: A

Cheers,
Brent

Thank you. I did the same thing you did to start the problem but I used $1000. Next I tried to use the percent change formula and that's where I went wrong. Could you conceptually explain why that wasn't the right choice? Also the part/whole = percent/100 formula would be the one to use here correct?
greenlight-admin's picture

The percent change formula will tell us the percent change from X to Y.

However, question #9 doesn't ask us to find the percent change from 2009 to 2010. Instead, we're asked the following:
The dollar amount of sales for 2009 was approximately WHAT PERCENT OF the dollar amount of sales for 2010?

In other words, we're asked "What percent of X is Y?"
So, we get: p% of X = Y (and then solve for p)

Or we can rephrase it as: "X/Y equals what percent?"
In which, we'll use: part/whole = percent/100
We get: 105/99.75 = p/100

Does that help?

Cheers,
Brent

In what questions can we apply this formula? I'm having a hard time with knowing when to use this or part/whole=%/100
greenlight-admin's picture

We can use that formula when we want to say that a certain quantity is comprised of a certain amount of one thing.

For example, if a team of 8 people is comprised of 3 males, then we can say: 3/8 = p/100
When we solve that equation for p, we'll get the percentage of the team members who are male.

Does that help?

Cheers,
Brent

Hi, for question http://www.urch.com/forums/gre/153587-please-help-detail-explanation.html I interpreted that the glass equals 8 oz. because that is how much a glass/cup measures to be as the standard unit of measurement. Then multiplied 20% by 8 oz to get the correct answer, 1.6 oz. Is there a reason why I would not want to do that to solve this? I do not understand how/why there was the need to calculate to get 8 oz.
greenlight-admin's picture

Question link: http://www.urch.com/forums/gre/153587-please-help-detail-explanation.html

The word "glass" doesn't have any specific measurement associated with it.
So, we can't say that the glass must hold 8 oz of liquid.

I hope that helps.

Cheers,
Brent

For: https://greprepclub.com/forum/joan-bought-a-calculator-at-a-discounted-price-that-was-30-p-3375.html ....got the answer fine with a calculator at end to do 107/.7 but would you calculate without a calculator?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/joan-bought-a-calculator-at-a-discounted-p...

I think you meant to say 105/0.7

Here's how I'd calculate 105/0.7 in my head"

105/0.7 = 105/(7/10)
= 105 x (10/7)
= 1050/7
= (700 + 350)/7
= 700/7 + 350/7
= 100 + 50
= 150

Cheers,
Brent

This is highly appreciated if could you explain Hilda's hat problem by using this formula (1+_ percentage change/ 100) x original?
Thanks
greenlight-admin's picture

You're referring to the question that begins at 3:45 of the above video.

Let $100 - the original price

20% INCREASE
New price = (1 + 20/100)($100)
= (1 + 0.2)($100)
= (1.2)($100)
= $120

15% DECREASE
Final price = (1 - 15/100)($120)
= (1 - 0.15)($120)
= (0.85)($120)
$102

Final price = $102

Cheers,
Brent

Yes, thank you! I knew there must be something basic I was missing.

Hiya, Brent!

I am confused about one of the study questions that's related to this topic. This one: https://greprepclub.com/forum/kathleen-s-weekly-salary-was-increased-by-8-percent-to-13649.html#p33761

I was able to set up the equation correctly, so I think this is more of an algebraic process question.
Here's your notes on the solution:

GIVEN: Kathleen’s weekly salary was increased by 8 percent to $237.60
We can write: x + (8% of x) = 237.60
Simplify: x + 0.08x = 237.60
Simplify: 1.08x = 237.60
Solve: x = 237.60/1.08 = 220

I am confused on how you went from this: Simplify: x + 0.08x = 237.60
To this: Simplify: 1.08x = 237.60

Specifically, where did the 1 come from? What was the step you did to simplify that?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/kathleen-s-weekly-salary-was-increased-by-...

Great question!

There's a bit of algebra going on here. The technique is called simplifying expressions, and it's covered later in the Algebra module here: https://www.greenlighttestprep.com/module/gre-algebra-and-equation-solvi...

Also, it's important to note that, if a term does not have a coefficient, then the coefficient is assumed to be 1.
For example, x = 1x and k² = 1k²

So, the equation: x + 0.08x = 237.60
Is the same as the equation: 1x + 0.08x = 237.60

-----------------------------
Aside: If two terms have the same variable, we can combine their coefficients. For example:
3x + 4x = 7x
8w² - 2w² = 6w²
k + k + k = 1k + 1k + 1k = 3k
----------------------------

Likewise, 1x + 0.08x = 237.60 can be simplified to 1.08x = 237.60

Does that help?

Cheers,
Brent

Hi Again Brent,

Working through some more practice, I see another thing I don’t understand with this question: https://greprepclub.com/forum/if-a-b-c-2-and-c-decreases-by-20-while-a-remains-5655.html

Here are your notes:

Given: a = b x c²
So, how about a = 2500, b = 100 and c = 5
We get: 2500 = 100 x 5²

c decreases by 20% while a remains constant
c becomes 4 and a stays at 2500
So, we get: 2500 = b x 4²
Simplify: 2500 = b x 16
Solve: 2500/16 = b
Simplify: 156.25 = b

So b INCREASES from 100 to 156.25
This represents an increase of 56.25%
Since the question asks us to round our answer to the nearest TENTH, the correct response is 56.3


What I am confused about is how reducing 5^2 by 20% gets you 4^2?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/if-a-b-c-2-and-c-decreases-by-20-while-a-r...

Be careful. We're not decreasing 5² by 20%; we're decreasing 5 by 20%. Here's why:

In my solution, I let c = 5

The question tells us that c decreases by 20%
20% of 5 is 1.
5 - 1 = 4
So, if we decrease 5 by 20%, we get 4.
In other words, c decreases from 5 to 4

This means, c² now equals 4², which equals 16

Does that help?

yup, thanks!

Hi brent! Can you please explain your answer in this: https://greprepclub.com/forum/quantitative-2290.html

Wouldn't the least value of a be 4.2 and the greatest value 8.4?

In which case 2a would have be be between 4.2(2) and 8.4(2) = 8.4 and 16.8.

In which case range would be between 4.7 and 13.1

Also I dont understand the extreme values you mention...how could the smallest value of a be 8.4999 ... wouldnt that be the largest value of a?
greenlight-admin's picture

Solution link: https://greprepclub.com/forum/quantitative-2290.html#p4753
Sorry, I mixed up the words GREATEST and SMALLEST in my original solution. I've edited to show the proper solution.

To answer your other questions, we know that a must be greater than 4.1.
While 4.2 is greater than 4.1, there are numbers smaller than 4.2 that are still greater than 4.1.
For example, 4.11 is greater than 4.1.
Likewise, 4.10000000000000000000000000000001 is greater than 4.1
In fact, if we keep adding more zeros, a can get super close to 4.1, which is why I just used 4.1 as a possible value of a.

Conversely, we know that a is less than 8.5.
So, a could equal 8.4999999999999999999999
If we keep adding more 9's, a can get super close to 8.5, which is why I just used 8.5 as a possible value of a.

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