Kesha paid a sales tax of x percent on her purchase. If the sales tax : Problem Solving (PS)

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16 Oct 2023, 14:46

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Let A be the purchase price excluding sales tax.

Given, $A(1+x) = $486$

Therefore, $A = \frac{$486}{(1+x)}$ (Equation 1)

Given, $Ax - \frac{A*2}{100} = $27$ (Equation 2)

Therefore substituting equation 1 in equation 2, $\frac{($486*x)}{(1+x)} - \frac{$486*2}{[(1+x)*100]} = $27$

On Solving, $x = 0.08 $
=> x = 8%

(A)

I was wondering if there are any other efficient ways of solving these types of percent problems which involve variables.

Anki111

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On such questions, back-solving can help. As formulating equation in a time crunch can lead to errors costing a question.
So, if you take each option and check, you will notice that the difference between 2% and the percentage options given, there is only one that can be an answer that is option A. 8%-2% i.e 6% of 486(it will be less than 486 excluding the sales tax), but still it will come 29.16. All other options, are way less than 27.

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24 Oct 2023, 22:40

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MrWhite

Kesha paid a sales tax of $x$ percent on her purchase. If the sales tax had only been $2$ percent, she would have paid $$27$ less in sales tax on her purchase. What was the value of $x$ if the total amount Kesha paid for her purchase, including sales tax, was $$486$?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 4

Assume that the price of the item without sales tax = $p$

if the total amount Kesha paid for her purchase, including sales tax, was $$486$

$p[1+\frac{x}{100}] = 486$

If the sales tax had only been $2$ percent, she would have paid $$27$ less in sales tax on her purchase

$p[1+\frac{x}{100}] - p[1+\frac{2}{100}] = 27$

$486 - p[1+\frac{2}{100}] = 27$

$486 - 27 = p[\frac{102}{100}]$

$p = \frac{459*100}{102} $

$ \frac{459*100}{102}[1+\frac{x}{100}] = 486$

$[1+\frac{x}{100}] = \frac{486 * 102}{459 * 100}$

$\frac{x}{100} = \frac{486 * 102}{459 * 100}-1$

$x = \frac{3672}{459}$

$x = 8$

Option A

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27 Dec 2023, 11:21

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MrWhite

Kesha paid a sales tax of $x$ percent on her purchase. If the sales tax had only been $2$ percent, she would have paid $$27$ less in sales tax on her purchase. What was the value of $x$ if the total amount Kesha paid for her purchase, including sales tax, was $$486$?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 4

Assume that the price of the item without sales tax = $p$

if the total amount Kesha paid for her purchase, including sales tax, was $$486$

$p[1+\frac{x}{100}] = 486$

If the sales tax had only been $2$ percent, she would have paid $$27$ less in sales tax on her purchase

$p[1+\frac{x}{100}] - p[1+\frac{2}{100}] = 27$

$486 - p[1+\frac{2}{100}] = 27$

$486 - 27 = p[\frac{102}{100}]$

$p = \frac{459*100}{102} $

$ \frac{459*100}{102}[1+\frac{x}{100}] = 486$

$[1+\frac{x}{100}] = \frac{486 * 102}{459 * 100}$

$\frac{x}{100} = \frac{486 * 102}{459 * 100}-1$

$x = \frac{3672}{459}$

$x = 8$

Option A

hi, how were you able to get to 3672 as the numerator- did you multiply them out or is there a faster way?

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MrWhite

Kesha paid a sales tax of $x$ percent on her purchase. If the sales tax had only been $2$ percent, she would have paid $$27$ less in sales tax on her purchase. What was the value of $x$ if the total amount Kesha paid for her purchase, including sales tax, was $$486$?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 4

Assume that the price of the item without sales tax = $p$

if the total amount Kesha paid for her purchase, including sales tax, was $$486$

$p[1+\frac{x}{100}] = 486$

If the sales tax had only been $2$ percent, she would have paid $$27$ less in sales tax on her purchase

$p[1+\frac{x}{100}] - p[1+\frac{2}{100}] = 27$

$486 - p[1+\frac{2}{100}] = 27$

$486 - 27 = p[\frac{102}{100}]$

$p = \frac{459*100}{102} $

$ \frac{459*100}{102}[1+\frac{x}{100}] = 486$

$[1+\frac{x}{100}] = \frac{486 * 102}{459 * 100}$

$\frac{x}{100} = \frac{486 * 102}{459 * 100}-1$

$x = \frac{3672}{459}$

$x = 8$

Option A

hi, how were you able to get to 3672 as the numerator- did you multiply them out or is there a faster way?

There is a way that avoids significant multiplication.

The first two statements are equivalent to:

2P/100 + 27 = PX/100

Since we're solving for X, eliminate P:

P(X-2)/100 = 27, so:

P = 2700/(X-2)

The third statement says:

P(1+ X/100) = 486.

Substituting for P:

(2700/(X-2))*((100+X)/100) =486

This reduces to:

486 = (2700+27X)/(X-2)

Expanding this out:

486X-972 = 2700+27X, or

459X = 3672 and

X = 3672/459

Now, doing this division precisely isn't required.

Multiplying 7 by 460 is 32 hundred something , so 7 is too small.

Multiplying 8 by 460 is 36 hundred something, which is close to the numerator, so 8 is the correct answer.

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MrWhite

Kesha paid a sales tax of $x$ percent on her purchase. If the sales tax had only been $2$ percent, she would have paid $$27$ less in sales tax on her purchase. What was the value of $x$ if the total amount Kesha paid for her purchase, including sales tax, was $$486$?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 4

Assume that the price of the item without sales tax = $p$

if the total amount Kesha paid for her purchase, including sales tax, was $$486$

$p[1+\frac{x}{100}] = 486$

If the sales tax had only been $2$ percent, she would have paid $$27$ less in sales tax on her purchase

$p[1+\frac{x}{100}] - p[1+\frac{2}{100}] = 27$

$486 - p[1+\frac{2}{100}] = 27$

$486 - 27 = p[\frac{102}{100}]$

$p = \frac{459*100}{102} $

$ \frac{459*100}{102}[1+\frac{x}{100}] = 486$

$[1+\frac{x}{100}] = \frac{486 * 102}{459 * 100}$

$\frac{x}{100} = \frac{486 * 102}{459 * 100}-1$

$x = \frac{3672}{459}$

$x = 8$

Option A

hi, how were you able to get to 3672 as the numerator- did you multiply them out or is there a faster way?

Hey gmatphobia , did you multiply 486∗102 ?

sayan640

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06 Mar 2024, 04:13

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Anki111

On such questions, back-solving can help. As formulating equation in a time crunch can lead to errors costing a question.
So, if you take each option and check, you will notice that the difference between 2% and the percentage options given, there is only one that can be an answer that is option A. 8%-2% i.e 6% of 486(it will be less than 486 excluding the sales tax), but still it will come 29.16. All other options, are way less than 27.

Can you please elaborate this short method ? gmatphobia GMATNinja Bunuel KarishmaB

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06 Mar 2024, 04:45

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Lets say , the purchase amount = P
P*(1+x/100) = 486---------- eqn (1)

P*x/100 - p*2/100 = 27-----------------------eqn(2)
Add p on both sides of eqn 2 ,

p + p*x/100 -p*2/100 = 27 +p
486 - 27 = p + 2*p/100 ...... ( Replacing P*(1+x/100) = 486 )
102p/100 = 459
p=450

Use it on eqn (2),
(p/100) * (x-2)=27
450/100 * (x-2) = 27
x = 8

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nisen20 That was amazing.

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MrWhite

Kesha paid a sales tax of x percent on her purchase. If the sales tax had only been 2 percent, she would have paid $27 less in sales tax on her purchase. What was the value of x if the total amount Kesha paid for her purchase, including sales tax, was $486?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 4

I got this in one of my mocks and here is how i approached it,

Given:
486 = C + (Ct/100)
486-27 = C+(2*C/100)
27 = C*(t-2)/100

459 = (100C+2C)/100
459 = 102C/100 = 51C/50
C = 459*50/51 = 9 * 50 = 450
C = 450

Now,
27 = C*(t-2)/100
27 = (450t – 900)/100
2700 = 450t – 900
450t = 3600
t = 360/45 = 40/5 = 8

t=tax, C=cost

C'est tout!

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30 Nov 2024, 23:41

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This is not a question where you wanna solve tedious equations.

So, I tried option elimination.

Of the options, let us choose the mdidle one first 6 %.
Now if x = 6%, x-2 = 4% and 4% of the sale price is equal to 27$. From this we can find the sale price = 675$, which is way greater thah 486. So we understand x=6 or any value <6 cannot be the answer.

We are now only left with 7 & 8.

Using the same logic with 7 , we find out that sale price is 540 $ , which is > than 486$. So the only option left is 8% which is the answer.

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Purchase amount including sale tax(x%)=486.
Purchase amount including sale tax if tax rate is 2%=[486][/1+X%]*1.02
As we know that difference is 27 and the equation would be 486-[486][/1+X%]*1.02 =27
x would be 8.

Kesha paid a sales tax of x percent on her purchase. If the sales tax

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