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edwin.que
At a certain store, the retail price of a coat was \(p\) percent less than its list price. If the sale price of the coat was \(r\) percent less than its retail price, then the sale price of the coat was what percent of its list price?

(1) \(p - r + \frac{pr}{100} = 10\)

(2) \(p + r - \frac{pr}{100} = 40\)


Assume that -
  • the retail price of the coat = \(R\)
  • the list price of the coat = \(L\)
  • the sale price of the coat = \(S\)

"...At a certain store, the retail price of a coat was \(p\) percent less than its list price..."

\(R = L(1-\frac{p}{100})\)

"...If the sale price of the coat was \(r\) percent less than its retail price..."

\(S = R(1-\frac{r}{100}) = L(1-\frac{p}{100})(1-\frac{r}{100})\)

Question: ...then the sale price of the coat was what percent of its list price...

\(\frac{S}{L} * 100\) = ?

Let's simplify this my multiplying \(\frac{100}{L}\) on both sides of the equation-

\(S = L(1-\frac{p}{100})(1-\frac{r}{100})\)

\(\frac{S}{L} * 100 = 100(1-\frac{p}{100})(1-\frac{r}{100})\)

\(\frac{S}{L} * 100 = 100(1-\frac{p}{100})(1-\frac{r}{100})\)

\(\frac{S}{L} * 100 = 100(1 - \frac{r}{100} - \frac{p}{100} + \frac{pr}{100^2})\)

\(\frac{S}{L} * 100 = 100 - r - p + \frac{pr}{100}\)

\(\frac{S}{L} * 100 = 100 - (r + p - \frac{pr}{100})\)

Let's start with Statement 2

Statement 2

(2) \(p + r - \frac{pr}{100} = 40\)

We are given the value of \(r + p - \frac{pr}{100}\), hence that's sufficient to answer the question.

Eliminate A, C, and E.

Statement 1

(1) \(p - r + \frac{pr}{100} = 10\)

Not sufficient, as we can multiple possible values of \(p\) and \(r\) satisfying the equation.

Option B
Hi @gmatophobia

How did you know that you have to multiply both sides by 100/L to arrive at the answer?
Can you pls elaborate on this

thanks
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Hi @gmatophobia

How did you know that you have to multiply both sides by 100/L to arrive at the answer?
Can you pls elaborate on this

thanks

Hey jack5397

From the premise we know that

\(S = L(1-\frac{p}{100})(1-\frac{r}{100})\) --- (1)

The questions asks us to find the value of

\(\frac{S}{L} * 100\) =?

Hence, we have multiplied equation (1) by \(100/L\) to arrive at the result.

Hope this helps.
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this one was tricky!!

but to solve it i just used a simple formula.

we know
Retail price = List price (1-p%)
Sale price = Retail price (1-r%)

we want \frac{Sale price }{ List price}

List price = L = Retail price/ (1-p%) = R/ (1-p%)

Sale price = S = Retail price (1-r%) = R (1-r%)

thus Sale price / List price = S/L = R (1-r%)/[ R/ (1-p%)] = (1-r%) (1-p%)

now we know that in the case of double increase or decrease we can easily calculate the total amount of decrease by using the formula:

-r%-p% + (pr/100)%

(1) this statement seems the formula! but the signs are not right!!! NOT SUFCICIENT

(2) this statement is our formula --> indeed if we just multiply both sides by -1 we get exactly the above -r%-p% + (pr/100)%

therefore: Sale price / List price = (1-r%) (1-p%) = -r%-p% + (pr/100)% = -40% SUFFICIENT
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KarishmaB
How to solve he problems with a lot of equations like these?
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Is there any effcient way to do this ?
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Sorry karishma but as shown in your website... the formula is not p+r+pr/100? Why is not important the sign before the term "pr/100"?
KarishmaB
edwin.que
At a certain store, the retail price of a coat was \(p\) percent less than its list price. If the sale price of the coat was \(r\) percent less than its retail price, then the sale price of the coat was what percent of its list price?

(1) \(p - r + \frac{pr}{100} = 10\)

(2) \(p + r - \frac{pr}{100} = 40\)

Attachment:
Coat.png

This is a 30 second question.

We are told that to the list price, a p% discount is applied (to get retail price) and then an r% discount is applied to get sale price. So 2 successive discounts are applied.

This means the overall discount is p + r - pr/100 (successive percentage changes that you must know for GMAT. It does appear in questions)

Statement 2 directly gives this value as 40% which means overall discount is 40% so sale price was 60% of list price.
Sufficient alone.

Statement 1 gives the relation between p and r but we cannot get p + r - pr/100 (overall discount percentage) from it. There is one equation with two variables. This is like saying that we need the value of x + y and we are given (x - y) = 10. Can we get (x+y)? No.

Answer (B)

Here is a post in which I have discussed this in successive percentage changes:
https://anaprep.com/arithmetic-successi ... e-changes/

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That's a great conceptual question.

The sign before the term pr/100 is very important as is the sign before any term in any formula.

Two Successive percentage changes a% and b% are equivalent to (a + b + ab/100) in percent terms. If it is mark up, the sign is positive (because value increases) and if it is a discount, sign is negative and value decreases. We plug in the formula based on whether it is a percentage increase or decrease. For decrease, we use a minus sign. So 2 decreases of 20% and 10% becomes an overall change of - 20 - 10 + (-20)-10)/100 = -28%

Now in this question, we are given that list price reduces by p% (say 20%) and sale price further reduces by r% (say 10%). So these p and r represent positive percentages in the formula they have given in statement 2.

We can say that a price CHANGES by a% in which case a can be -20 or we can say that the price REDUCES by p% so here p = 20.

So in our formula, we would write overall percentage CHANGE as (-p - r + pr/100)% or we could say that price overall REDUCES by (p + r - pr/100)%


That is why statement 2 is correct. If overall reduction in price is of 40% then we know that sale price was 60% of its list price.



Pif96
Sorry karishma but as shown in your website... the formula is not p+r+pr/100? Why is not important the sign before the term "pr/100"?
KarishmaB
edwin.que
At a certain store, the retail price of a coat was \(p\) percent less than its list price. If the sale price of the coat was \(r\) percent less than its retail price, then the sale price of the coat was what percent of its list price?

(1) \(p - r + \frac{pr}{100} = 10\)

(2) \(p + r - \frac{pr}{100} = 40\)

Attachment:
Coat.png

This is a 30 second question.

We are told that to the list price, a p% discount is applied (to get retail price) and then an r% discount is applied to get sale price. So 2 successive discounts are applied.

This means the overall discount is p + r - pr/100 (successive percentage changes that you must know for GMAT. It does appear in questions)

Statement 2 directly gives this value as 40% which means overall discount is 40% so sale price was 60% of list price.
Sufficient alone.

Statement 1 gives the relation between p and r but we cannot get p + r - pr/100 (overall discount percentage) from it. There is one equation with two variables. This is like saying that we need the value of x + y and we are given (x - y) = 10. Can we get (x+y)? No.

Answer (B)

Here is a post in which I have discussed this in successive percentage changes:
https://anaprep.com/arithmetic-successi ... e-changes/

MBAToronto2024
anirchat
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Hi KarishmaB,

Let's say i take list price as 100 for ease.
In that case retail price= 100-p
and sale price= (100-p)(1-r/100)

Simplifying the eqn we get Sale price= 100-p-r+pr/100

In that case isnt statement A sufficient as well?

It gives us the value of p-r+pr/100 so that sale price= 100-10 = 90

Please help out

KarishmaB
edwin.que
At a certain store, the retail price of a coat was \(p\) percent less than its list price. If the sale price of the coat was \(r\) percent less than its retail price, then the sale price of the coat was what percent of its list price?

(1) \(p - r + \frac{pr}{100} = 10\)

(2) \(p + r - \frac{pr}{100} = 40\)

Attachment:
Coat.png

This is a 30 second question.

We are told that to the list price, a p% discount is applied (to get retail price) and then an r% discount is applied to get sale price. So 2 successive discounts are applied.

This means the overall discount is p + r - pr/100 (successive percentage changes that you must know for GMAT. It does appear in questions)

Statement 2 directly gives this value as 40% which means overall discount is 40% so sale price was 60% of list price.
Sufficient alone.

Statement 1 gives the relation between p and r but we cannot get p + r - pr/100 (overall discount percentage) from it. There is one equation with two variables. This is like saying that we need the value of x + y and we are given (x - y) = 10. Can we get (x+y)? No.

Answer (B)
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That's not correct.

100-p-r+pr/100 = 100 - (p + r - pr/100)

Note the negative signs. Statement 1 gives the value of (p - r + pr/100)



Jeeya
Hi KarishmaB,

Let's say i take list price as 100 for ease.
In that case retail price= 100-p
and sale price= (100-p)(1-r/100)

Simplifying the eqn we get Sale price= 100-p-r+pr/100

In that case isnt statement A sufficient as well?

It gives us the value of p-r+pr/100 so that sale price= 100-10 = 90

Please help out

KarishmaB
edwin.que
At a certain store, the retail price of a coat was \(p\) percent less than its list price. If the sale price of the coat was \(r\) percent less than its retail price, then the sale price of the coat was what percent of its list price?

(1) \(p - r + \frac{pr}{100} = 10\)

(2) \(p + r - \frac{pr}{100} = 40\)

Attachment:
Coat.png

This is a 30 second question.

We are told that to the list price, a p% discount is applied (to get retail price) and then an r% discount is applied to get sale price. So 2 successive discounts are applied.

This means the overall discount is p + r - pr/100 (successive percentage changes that you must know for GMAT. It does appear in questions)

Statement 2 directly gives this value as 40% which means overall discount is 40% so sale price was 60% of list price.
Sufficient alone.

Statement 1 gives the relation between p and r but we cannot get p + r - pr/100 (overall discount percentage) from it. There is one equation with two variables. This is like saying that we need the value of x + y and we are given (x - y) = 10. Can we get (x+y)? No.

Answer (B)
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QUICK PICK: Seeing this question clearly tells us that answer must be either option A or Option B (only for those who see such question when they are running short of time. It's called intelligent elimination of options

Let's do it mathematically now

RP = (1-p/100)LP
Sale Price = (1-r/100)RP = (1-r/100)(1-p/100)LP = [1-{p+r-(pr/100)}/100]LP

So, we need [1-{p+r-(pr/100)}/100]

Question: [1-{p+r-(pr/100)}] = ?

Statement 1:\(p - r + \frac{pr}{100} = 10\)
Not what we want hence
NOT SUFFICIENT

Statement 1:\(p + r - \frac{pr}{100} = 40\)
EXACTLY what we want hence
SUFFICIENT

Answer: Option B


Related Video:


edwin.que
At a certain store, the retail price of a coat was \(p\) percent less than its list price. If the sale price of the coat was \(r\) percent less than its retail price, then the sale price of the coat was what percent of its list price?

(1) \(p - r + \frac{pr}{100} = 10\)

(2) \(p + r - \frac{pr}{100} = 40\)

Attachment:
Coat.png
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In order to answer this question what do we need to know? We need to know the sale price divided by the list price. (And what they’re really testing is how well you know the “algebraic identity” of consecutive percent changes. See below.)

Let's start by writing equations.

Suppose the list price is equal to L, the retail price is equal to R, and the sale price is equal to S.

R = L * (1 - p/100)
S = L * (1 - p/100) * (1 - r/100)

We want to find S/L, which means we want to find (1 - p/100) * (1 - r/100)

Let’s use “FOIL” from algebra class to re-write that value (an excellent exercise – go ahead and give it a try on a piece of paper) in a way that’s more similar to what we see in statements 1 and 2.

(1 - p/100) * (1 - r/100) = 1 – r/100 – p/100 + rp/10,000

This equation, the algebraic identity of consecutive percent changes, is important to know for the GMAT.

So the question behind the question – what we need to know in order to answer this question – is, “What is the value of (1 – r/100 – p/100 + rp/10,000)”?

That can be re-written as

(1 – (r/100 + p/100 - rp/10,000) ), so we could also phrase our question as

“What is the value of (r/100 + p/100 - rp/10,000)?”


Statement 1:

p – r + pr/100 = 10

Divide both sides by 100 and we have

p/100 – r/100 + pr/10,000 = 10/100. But p/100 – r/100 + pr/10,000 is not the same as either expression written above.

NOT SUFFICIENT


Statement 2:

p + r - pr/100 = 40

Divide both sides by 100 and we have

p/100 + r/100 – pr/10,000 = 40/100

The expression p/100 + r/100 – pr/10,000 is equal to our expression above r/100 + p/100 - rp/10,000.

SUFFICIENT
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