GMAT - Percentage

Question 1

When there are only two successive percentage changes, we can derive a formula. In some cases, the formula makes the solution very simple

This formula is used only when there are two successive percentage changes and the percentages are easy to work with e.g. 15% and 25%, -10% and – 30% etc.
.

Answer 1

When a number, N, changes by x% and then changes again by y%, we do the following to find the new number:

New number = N * (1 + x/100) * (1 + y/100)

Now, (1 + x/100) * (1 + y/100) = 1 + x/100 + y/100 + xy/10000

If we say that x + y + xy/100 = z, then (1 + x/100) * (1 + y/100) = 1 + z/100

Here, z is the effective percentage change when a number is changed successively by two percentage changes. Let’s take another example to see the formula in action:

Example 3:

A city’s population was 10,000 at the end of 2008. In 2009, it increased by 20% and in 2010, it decreased by 10%. What was the city’s population at the end of 2010?

x% = 20%

y% = – 10% (Notice the negative sign here because this is a decrease)

Effective percentage change = x + y + xy/100 = 20 + (– 10) + 20*(-10)/100 = 8%

Population at the end of 2010 = 10,000 * (108/100) = 10800

Note: When the percentage is a decrease, a negative sign is used as shown above.

Question 2

Month Change in sales from previous month
February. +10%

March −15%

April +20%

May −10%

June +5%

The table above shows the percent of change from the previous month in Company X’s sales for February through June of last year. A positive percent indicates that Company X’s sales for that month increased from the sales for the previous month, and a negative percent indicates that Company X’s sales for that month decreased from the sales for the previous month. For which month were the sales closest to the sales in January?

Answer 2

May

When the percentage changes are numbers such as 10, 15, 20 etc, it is easier to use the formula:

Total change = a + b + ab/100

March total change = 10 - 15 -1015/100 = -6.5
April total change = -6.5 + 20 -6.520/100 = 12 (approx)
May total change = -10 + 12 -10*12/100 = 0.6 (approx)

Next change is increase of 5% so it will increase the total change. Hence May has the smallest overall change from Jan of about 0.6%.

Question 3

A dealer offers a cash discount of 20%. Further, a customer bargains and receives 20 articles for the price of 15 articles. The dealer still makes a profit of 20%. How much percent above the cost price were his articles marked?

a) 100%
b) 80%
c) 75%
d) 66+2/3%
e) 50%

Answer 3

Ans = 100%

(1+Mark Up%)∗(1–Discount%)=(1+Profit%)

A dealer offers a cash discount of 20%. Further, a customer bargains and receives 20 articles for the price of 15 articles. The dealer still makes a profit of 20%. How much percent above the cost price were his articles marked?

a) 100%
b) 80%
c) 75%
d) 66+2/3%
e) 50%

This question involves two discounts:

1. the straight 20% off
2. discount offered by selling 20 articles for the price of 15 – a discount of cost price of 5 articles on 20 articles i.e. a discount of 5/20 = 25%

Using the formula given above:

(1+m100)(1–20100)(1–25100)=(1+20100)(1+m100)(1–20100)(1–25100)=(1+20100)
m = 100

Therefore, the mark up was 100%.

Question 4

Compound Interest - Can be treated like successive percentage changes

Q1 : A bank launched a new financial instrument called VDeposit. A VDeposit offers you variable rate of compound interest in accordance with the current market rate. Ethan deposited $8000 in a VDeposit. If he gets interest rates of 10% in the first two years and 12.5% in the third year, what is the total amount at the end of 3 years?

Q2: Jolene entered an 18-month investment contract that guarantees to pay 2 percent interest at the end of 6 months, another 3 percent interest at the end of 12 months, and 4 percent interest at the end of the 18 month contract. If each interest payment is reinvested in the contract, and Jolene invested $10,000 initially, what will be the total amount of interest paid during the 18-month contract?

A. $506.00
B. $726.24
C. $900.00
D. $920.24
E. $926.24

Answer 4

Ans = $10,890

8000(11/10)(11/10)(9/8) = 1000(11/10)(11/10)(9)

Q2 ANS = E

When rate of interest is R1%, R2% and R3% for 1st yr, 2nd yr and 3rd yr respectively, then

Amount=Principal[(1+R1100)(1+R2100)(1+R3100)]Amount=Principal[(1+R1100)(1+R2100)(1+R3100)]

Applying the same formula here in this problem:

Amount=10000[(1+2100)(1+3100)(1+4100)]Amount=10000[(1+2100)(1+3100)(1+4100)]

= 10000∗102100∗103100∗10410010000∗102100∗103100∗104100

= 102∗103∗104100102∗103∗104100

= 926.24 = OPTION E [Ans]
+1

Question 5

CI 3 years

Bob invested one half of his savings in a bond that paid simple interest for 3 years and received $825 as interest. He invested the remaining in a bond that paid compound interest (compounded annually) for the same 3 years at the same rate of interest and received $1001 as interest. What was the annual rate of interest?

(A) 5%
(B) 10%
(C) 12%
(D) 15%
(E) 20%

Answer 5

Ans = 20 %

Simple Interest for three years = $825

So simple interest per year = 825/3 = $275

But in case of compound interest, you earn an extra $1001 – $825 = $176

What all is included in this extra $176? This is the extra interest earned by compounding.

This is R% of interest of Year1 + R% of total interest accumulated in Year2

This is R% of 275 + R% of (275 + 275 + R% of 275) = 176

(R100)∗[825+(R100)∗275]=176(R100)∗[825+(R100)∗275]=176

Assuming R100=xR100=x to make the equation easier,

275x2+825x–176=0275x2+825x–176=0

25x2+75x–16=025x2+75x–16=0

25x2+80x–5x–16=025x2+80x–5x–16=0

5x(5x+16)–1(5x+16)=05x(5x+16)–1(5x+16)=0

x=15x=15 or −165−165

Ignore the negative value to get R100=15R100=15 or R=20R=20.

Question 6

Mark deposited $D in a scheme offering 5% simple interest per annum. Tetha deposited $D in a scheme offering 5% compound interest per annum. At the end of second year, Tetha had earned a total of $2.50 more than Mark. What is the value of D?

Answer 6

Ans = 1000

Till the end of first year, simple interest and compound interest cases are exactly the same. The difference comes in at the end of second year when compound interest offers interest on previous year’s interest too. $2.50 is 5% interest earned in the second year on first year’s interest.

2.5 = (5/100) * I

I = $50

So interest earned in the first year is $50, which is 5% of the deposited amount D

50 = (5/100)*D

D = $1000

Question 7

Difference between Simple interest and compound interest

Answer 7

We saw that simple and compound interest (compounded annually) in the first year is the same. In the second year, the only difference is that in compound interest, you earn interest on previous year’s interest too. Hence, the total two year interest in compound interest exceeds the two year interest in case of simple interest by an amount which is interest on year 1 interest.

Question 8

Formula for successive percentage changes

Answer 8

When a number, N, changes by x% and then changes again by y%, we do the following to find the new number:

New number =N∗(1+x100)∗(1+y100)=N∗(1+x/100)∗(1+y/100)

Now, (1+x/100)∗(1+y/100)=1+x/100+y/100+xy/10000

If we say that x+y+xy/100=z
then 1+z/100

Question 9

when to use successive percentage formula

when not to use successive percentage formula

Answer 9

This formula is used only when there are two successive percentage changes and the percentages are easy to work with e.g. 15% and 25%, -10% and – 30% etc.

With more than two successive percentage changes or trickier percentage values e.g. 11.11% and 18.18%, 9.09% and 6.25% etc, stick to the method shown above.

Question 10

When two items are sold at the same selling price, one at a profit of x% and the other at a loss of x%, there is an overall loss.

Application -

John bought 2 shares and sold them for $96 each. If he had a profit of 20% on the sale of one of the shares but a loss of 20% on the sale of the other share, then on the sale of both shares John had
(A) a profit of $10
(B) a profit of $8
(C) a loss of $8
(D) a loss of $10
(E) neither a profit nor a loss

Answer 10

The loss% = (x^2/100)%

Ans = 8

Note that the question would have been straight forward had the COST price been the same, say $100. A 20% profit would mean a gain of $20 and a 20% loss would mean a loss of $20. Overall, there would have been no profit no loss.

Here the two shares are sold at the same SALE price. One at a profit of 20% on cost price which must be lower than the sale price (to get a profit) and the other at a loss of 20% on cost price which must be higher than the sale price (to get a loss). 20% of a lower amount will be less in dollar terms and hence overall, there will be a loss.

The loss % =(20)2100%=4%=(20)2100%=4%.

But we need the amount of loss, not the percentage of loss.

Total Sale price of the two shares = 2∗96=$1922∗96=$192

Since there is a loss of 4%, the 96% of the total cost price must be the total sale price

(96100)∗Cost Price=Sale Price(96100)∗Cost Price=Sale Price

Cost Price=$200Cost Price=$200

Loss=$200–$192=$8Loss=$200–$192=$8

Question 11

Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex deposited an additional x dollars into the account. If there were no other transactions and if the account contained w dollars at the end of two years, which of the following expresses x in terms of w ?

A. w1+1.08w1+1.08

B. w1.08+1.16w1.08+1.16

C. w1.16+1.24w1.16+1.24

D. w1.08+1.082w1.08+1.082

E. w1.082+1.082

Answer 11

ans = D

Question 12

On the first of the year, James invested x dollars at Proudstar bank in an account that yields 2% in interest every quarter year. At the end of the year, during which he made no additional deposits or withdrawals, he had and dollars in the account. If James had invested the same amount in an account which pays interest on a yearly basis, what must the interest rate be for James to have y dollars at the end of the year?

A. 2.04%
B. 6.12%
C. 8%
D. 8.25%
E. 10%

Answer 12

My friend, in most standard problems, the interest rate given is an ANNUAL interest rate, and for compounding quarterly, we have to divide it by four, as that formula does. BUT, in this problem we are told:

On the first of the year, James invested x dollars at Proudstar bank in an account that yields 2% in interest every quarter year .

So, this problem is following a different pattern — it is not giving us an ANNUAL interest rate that needs to be divided by four. It is giving us a QUARTERLY interest rate.

So, every increase of 2% means we multiply x by the multiplier 1.02. The initial amount x gets multiplied by this multiply four times, one for each quarter, so . . .

y = x(1.02)^4 = (1.08243216)x ====> effective interest = 8.2432%

That’s how you’d get the exact answer with a calculator, but of course you don’t have a calculator available on GMAT PS questions. Think about it this way. With simple interest, 2% a quarter would add up to 8% annually. With compound interest, where you get interest on your interest, you will do a little better than you would with simple interest, so the answer should be something slightly above 8%. That leads us to . . .

Question 13

Louie takes out a three-month loan of $1000. The lender charges him 10% interest per month compunded monthly. The terms of the loan state that Louie must repay the loan in three equal monthly payments. To the nearest dollar, how much does Louie have to pay each month?

A. 333
B. 383
C. 402
D. 433
E. 483

Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over $100 in interest within 6 months?

A. 1500
B. 1750
C. 2000
D. 2500
E. 3000

Answer 13

Q1 - Ans = C

this is because you are paying off in the third and last months. This is assuming the interest rate is calculated at the end of the month. So it is assumed you paid off the balance at the end of third month so 0 balance. Like CC statements - if you didnt pay off your statement by end of month you get charged interest - you dont get charged interest throughout.

Q2 - ANS - D

Annual rate of 8% compounded quarterly is approximately 4% in 6 months (a bit more).

x*0.04=100 –> x=2500.

Answer: D.

Question 14

Be careful of wording -

interest at an annual rate of 8 percent compounded semiannually

Question 15

Diana invested $61,293 in an account with a fixed annual percent of interest, compounding quarterly. At the end of five full years, she had $76,662.25 in principal plus interest. Approximately what was the annual percent rate of interest for this account?

A. 1.2%
B. 4.5%
C. 10%
D. 18%
E. 25.2%

Answer 15

Ans = B

We have some ugly numbers and are asked to find approximate percent, so we can approximate and use shortcuts.

~$15,000 of interest in 5 years –> $3,000 per year –> 3,000/60,000*100 = 5%.

Question 16

A retail item is offered at a discount of p percent (where p > 10), with a 5% state sales tax assessed on the discounted purchase price. If the state sales tax were not assessed, what percent discount from the item’s original retail price, in terms of p, would result in the same final price?

A. p+5/1.05
B. p/1.05 +5
C. 1.05p-5
D. p-5/1.05
E. 1.05(p – 5)

Answer 16

Ans = C

The original retail price and the value of p are undetermined, so both can be selected arbitrarily.

Let the original retail price be $100, and let p = 20. Then the discounted price is 20% off $100, or $80. Five percent of 80 is 4, so adding the 5% sales tax gives a final price of $84.

If no sales tax is assessed, then a final price of $84 is equivalent to a discount of $16, or 16% off the original retail price.

Check the answer choices:

(A)   25/1.05 = slightly less than 25.
(B)   20/1.05 + 5 = (slightly less than 20) + 5 = slightly less than 25.
(C)   1.05(20) – 5 = 21 – 5 = 16.
(D)   15/1.05 = less than 15.
(E)    1.05(15) = 15.75.

Only choice (C) works, so the correct answer is (C).

This problem can also be solved algebraically.

Let a stand for the required discount (without sales tax), and R the original retail price. Then the result of decreasing R by p percent, and then adding 5% sales tax, must be the same as the result of decreasing R by a percent:

        R(100 – p)(1.05) = R(100 – a)

Cancel R and distribute:

        105 – 1.05p = 100 – a
        5 – 1.05p = –a
        –5 + 1.05p = a

The correct answer is (C).

Question 17

Jack and Mark both received hourly wage increases of 6 percent. After the increases, Jack’ hourly wage was how many dollars per hour more than Mark’s?

(1) Before the wage increases, Jack’s hourly wage is $5 per hour more than Mark’s
(2) Before the wage increases, the ratio of the Jack’s hourly wage to Mark’s hourly wage is 4 to 3

Answer 17

(1) Before the wage increases, Jack’s hourly wage is $5 per hour more than Mark’s –> no need for algebra here: as wages increase by 6% then the difference between them also increases by the same 6% –> new difference 5*1.06. Sufficient.
(2) Before the wage increases, the ratio of the Jack’s hourly wage to Mark’s hourly wage is 4 to 3 –> just ratio is not enough to get the value of new difference, as for different hourly wages we’ll get different result: bigger the wages the bigger the difference will be. Not sufficient.

Question 18

Martha bought an armchair and a coffee table at an action and sold both items at her store. Her gross profit from the purchase and sale of the armchair was what percentage greater than her gross profit from the purchase and sale of the coffee table?

(1) Martha paid 10% more for armchair than for the coffee table.
(2) Martha sold the armchair for 20% more than she sold the coffee table.

Answer 18

ans = E

(1) Martha paid 10 percent more for the armchair than for the coffee table.

This means Ca = 1.1Cc

(2) Martha sold the armchair for 20 percent more than she sold the coffee table.

This means Sa = 1.2Sc

Both statements together give us (1.2Sc - 1.1Cc)/(Sc - Cc) which is still not enough. Hence answer is (E).

In case, instead, we had (1.1Sc - 1.1Cc)/(Sc - Cc) or (1.2Sc - 1.2Cc)/(Sc - Cc) etc, we could have taken the common factor out and cancelled it and we would have got 1.1 - 1 = 10% or 20% etc. In that case, answer would have been (C).

In essence, if the cost price of armchair is more than the cost price of coffee table by some percent x and if the selling price is also more by the same percent x, then the gross profit on the armchair would also be more by the same percent x.

Question 19

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.

Answer 19

Ans = A

Given: {Income} = {salary} + {commission}. Question basically asks: is {salary} > {commission}?

(1) If the amount of the commission had been 30 percent higher, the salesman’s income would have been 10 percent higher last year:
1. 1({salary} + {commission}) = {salary} + 1.3{commission};

{salary} = 2{commission};

Therefore, {salary} > {commission}. Sufficient.

(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:

|{salary} - {commission}| = 0.5{salary}, notice that {salary} - {commission} is in absolute value sign ||, meaning that we can have two cases:

A. {salary} - {commission} = 0.5{salary};

0.5{salary} = {commission};

{salary} > {commission}, thus the answer would be YES;

Or:
B. {commission} - {salary} = 0.5{salary};

1.5{salary} = {commission};

{salary} < {commission}, thus the answer would be No.

Two different answers. Not sufficient.

Answer: A.

Question 20

Is x% of x% of y equal to x% less than y ?

(1) x(x + 100) = 10,000
(2) y(y + 1) = 1

Answer 20

I just cancelled out Y which was wrong so understand why that should not be done

https://gmatclub.com/forum/is-x-of-x-of-y-equal-to-x-less-than-y-1-x-x-101953.html

Question 21

Bobby bought two shares of stock, which he sold for $96 each. If he had a profit of 20 percent on the sale of one of the shares but a loss of 20 percent on the sale of the other share, then on the sale of both shares combined Bobby had

(A) a profit of $10
(B) a profit of $8
(C) a loss of $8
(D) a loss of $10
(E) neither a profit nor a loss

Answer 21

I did not read the language of the question properly, I thought it was asking for percentage but it was asking for the actual value

Question 22

Angela’s grade was in the 90th percentile out of 80 grades in her class. In another class of 100 students there were 19 grades higher than Angela’s. If nobody had Angela’s grade, then Angela was what percentile of the two classes combined?

a. 72
b. 80
c. 81
d. 85
e. 92

Answer 22

She is in 90th percentile in one class and 81st in the other. We need to find her ‘average percentile’. The average will lie between 81 and 90 so it has to be 85!

If we had closer answer options e.g. 84, 86 etc, we would need to calculate the weighted average.
Using weights as 80 and 100 respectively, you can find their average using weighted average formula which is as given below:
Cavg=(C1∗W1+C2∗W2)/(W1+W2)

Cavg=(90∗80+81∗100)/180. = 85

Question 23

All votes cast in a recent presidential election were for either the incumbent or thechallenger. The challenger received 5.4 million votes and the incumbent received 5million. If after a recount of the votes and the addition of previously uncountedabsentee ballots, the incumbent had 5.2 million votes while the challenger had 5.4million, then the percentage of the total number of votes that were for the challenger

A. decreased by approximately 10 %
B. decreased approximately 1%
C. neither increased nor decreased
D. increased approximately 1%
E. increased approximately 2%

Answer 23

Ans = B

Before a recount the percentage of the total number of votes that were for the challenger was 5.4/ (5+5.4)∗100 = 5.4/10.4∗100;
After the recount the percentage of the total number of votes that were for the challenger was 5.4/(5.2+5.4)∗100=5.4/10.6∗100;

Since the total # of votes increased (denominator increased from 10.4 to 10.6) and the # of votes for the challenger remained the same then the percentage of the total number of votes that were for the challenger clearly decreased, though since the increase in the total # of votes were minimal then the decrease in the percentage of the total number of votes that were for the challenger would also be minimal.
Only Denominator increase from 10.4 to 10.6 (around 2% increase) = so ratio decreases (around 1%)

Only answer choice B fits.

Question 24

On average, the bottle-nosed dolphin comes up for air once every two minutes; the beluga whale, a close relative, comes up for air on average once every five minutes. The number of times a bottle-nosed dolphin would come up for air in a 24 hour period is approximately what percent greater than the number of times a beluga whale would come up for air in that same period?

A. 50%
B. 100%
C. 150%
D. 200%
E. 250%

Answer 24

Dolphin once in 2 min;
Beluga once in 5 min;
So, dolphin comes up 2.5 times frequently than beluga, which is 150% (5-2)/2*100.

Answer: C.

Question 25

A certain portfolio consisted of 5 stocks, priced at $20, $35, $40, $45 and $70, respectively. On a given day, the price of one stock increased by 15%, while the price of another decreased by 35% and the prices of the remaining three remained constant. If the average price of a stock in the portfolio rose by approximately 2%, which of the following could be the prices of the shares that remained constant?

A) 20, 35, 70

B) 20, 45, 70

C) 20, 35, 40

D) 35, 40, 70

E) 35, 40, 45

Answer 25

Use logic and the calculations involved will be negligible.
Since we have been given that there is an overall increase, the increase should be greater than the decrease. So
15% of A > 35% of B

Now think that 15% of A will be equal to 30% of B if A is twice of B. But 15% of A is greater than 30% of B so A must be greater than twice of B. In fact 15% of A is greater than 35% of B so A must be substantially greater than twice of B. So B has to be 20 because we have values more than twice of 20 (which are 45 and 70). We don’t have any values which are more than twice of any other given number (30, 35, 40, 70).

A can be 45 or 70. I would bet on 70 since A has be substantially greater than twice of 20.
Even if I want to confirm, 10% of 20 is 2 so 30% of 20 is 6. 5% of 20 is 1 so 35% of 20 is 7.
15% of A has to be greater than 7. Only 70 satisfies this since 10% of 70 is 7.

Question 26

An investment has been growing at a fixed annual rate of 20% since it was first made; no portion of the investment has been withdrawn, and all interest has been reinvested. How much is the investment now worth?

(1)  The value of the investment has increased by 44% since it was first made.
(2)  If one year ago $600 had been withdrawn, today the investment would be worth 12% less than it is actually now worth.

Answer 26

ans = b