Test: Percentages- 3 - UPSC MCQ

Percentage problems involving unspecified amounts can usually be solved more easily by using the number 100. If Arthur's fortune was originally $100, each of his children received $20.

Let's see what happened to each $20 investment in the first year:

Alice:  $20 + $10 profit = $30
Bob:  $20 + $10 profit = $30
Carol:  $20 + $10 profit = $30
Dave:  $20 - $8 loss = $12
Errol:  $20 - $8 loss = $12

We continue on with our new amounts in the second year:

Alice:  $30 + $3 profit = $33
Bob:  $30 + $3 profit = $33
Carol:  $30 - $18 loss = $12
Dave:  $12 + $3 profit = $15
Errol:  $12 - $12 = 0

At the end of two years, $33 + $33 + $12 + $15 = $93 of the original $100 remains.

The correct answer is A. 

This is a weighted average problem; we cannot simply average the percentage of silver cars for the two batches because each batch has a different number of cars.
The car dealership currently has 40 cars, 30% of which are silver. It receives 80 new cars, 60% of which are silver (the 40% figure given in the problem refers to cars which are not silver). Note that the first batch represents 1/3 of the total cars and the second batch represents 2/3 of the total cars. Put differently, in the new total group there is 1 first-batch car for every 2 second-batch cars.

We can calculate the weighted average, weighting each percent according to the ratio of the number of cars represented by that percent:
Weighted average = 

Alternatively, you can calculate the actual number of silver cars and divide by the total number of cars. 40(0.3) + 80(0.6) = 12 + 48 = 60. 60/120 = 50%.
The correct answer is E.

Notice that Paul’s income is expressed as a percentage of Rex’s and that the other two incomes are expressed as a percent of Paul’s. Let’s assign a value of $100 to Rex’s income. Paul’s income is 40% less than Rex's income, so (0.6)($100) = $60. Quentin’s income is 20% less than Paul's income, so (0.8)($60) = $48. Sam’s income is 40% less than Paul's income, so (0.6)($60) = $36. If Rex gives 60% of his income, or $60, to Sam, and 40% of his income, or $40, to Quentin, then: Sam would have $36 + $60 = $96 and Quentin would have $48 + $40 = $88. Quentin’s income would now be $88/$96 = 11/12 that of Sam's.
The correct answer is A.

Let’s denote the formula for the money spent on computers as pq = b, where 
p = price of computers
q = quantity of computers
b = budget 

We can solve a percent question that doesn’t involve actual values by using smart numbers. Let’s assign a smart number of 1000 to last year’s computer budget (b) and a smart number 100 to last year’s computer price (p).  1000 and 100 are easy numbers to take a percent of.

This year’s budget will equal 1000 × 1.6 = 1600 
This year’s computer price will equal 100 × 1.2 = 120 

Now we can calculate the number of computers purchased each year, q = b/p 
Number of computers purchased last year = 1000/100 = 10 
Number of computers purchased this year = 1600/120 = 13 1/3 (while 1/3 of a computer doesn’t make sense it won’t affect the calculation)

The question is asking for the percent increase in quantity from last year to this year =


 

This question could also have been solved algebraically by converting the percent increases into fractions. 

Last year: pq = b, so q = b/p 
This year: (6/5)(p)(x) = (8/5)b 
If we solve for x (this year's quantity), we get x = (8/5)(5/6)b/p  or (4/3)b/p 
If this year's quantity is 4/3 of last year's quantity (b/p), this represents a 33 1/3% increase. 

The correct answer is A

This problem can be solved most easily with the help of smart numbers. With problems involving percentages, 100 is typically the ‘smartest’ of the smart numbers. 

If we assume that today’s population is 100, next year it would be 1.1 × 100 = 110, and the following year it would be 1.1 × 110 = 121. If this is double the population of one year ago, the population at that time must have been 0.5 × 121 = 60.5. Because the problem seeks the “closest” answer choice, we can round 60.5 to 60. 

In this scenario, the population has increased from 60 to 100 over the last year, a net increase of 40 residents. To determine the percentage increase over the last year, divide the net increase by the initial population: 40/60 = 4/6 = 2/3, or roughly 67%. 

For those who prefer the algebraic approach: let the current population equal p. Next year the population will equal 1.1p, and the following year it will equal 1.1 × 1.1p = 1.21p. Because the question asks for the closest answer choice, we can simplify our algebra by rounding 1.21p to 1.2p. Half of 1.2p equals 0.6p. The population increase would be equal to 0.4p/0.6p = 0.4/0.6 = 2/3, or roughly 67%. 

The correct answer is D
 

To solve this problem, first find the wholesale price of the shirt, then compute the price required for a 100% markup, then subtract the $45 initial retail price to get the required increase. 

Let x equal the wholesale price of the shirt. The retailer marked up the wholesale price by 80% so the initial retail price is x + (80% of x). The following equation expresses the relationship mathematically:

x + 0.80x = 45
1.8x = 45
x = 45/1.8
x = 450/18
x = 25

Since the wholesale price is $25, the price for a 100% markup is $50. Therefore the retailer needs to increase the $45 initial retail price by $5 to achieve a 100% markup.

The correct answer is E. 

We can solve this as a VIC (Variable In answer Choices) and plug in values for x and r. 

Since there are 3 people, the taxi driver will charge them 30 cents per mile.  
Since they want to travel 20 miles, the total charge (no discount) would be (30)(20) = 600.
With a 50% discount, the total charge will be 300 cents or 3 dollars.
If we plug r = 10 and x = 20 into the answer choices, the only answer that yields 3 dollars is D.
The correct answer is D.

Bob put 20 gallons of gasohol into his car's gas tank, consisting of 5% ethanol and 95% gasoline. Chemical Mixture questions can be solved by using a mixture chart. 

This chart indicates that there is 1 gallon of ethanol out of the full 20 gallons, since 5% of 20 gallons is 1 gallon.

Now we want to add x gallons of ethanol to raise it to a 10% ethanol mixture. We can use another mixture chart to represent the altered solution.

Therefore, the following equation can be used to solve for x:

The correct answer is C.

Noting that 65% is very close to 2/3, we may approximate the original expression as follows:

1/3 + 0.4 + 65%     Original expression

1/3 + 0.4 + 2/3       Close approximation

1 + 0.4

1.4 

The correct answer is D.

First, let’s find the initial amount of water in the tank: 
Total mixture in the tank =1/4 × (capacity of the tank) = (1/4) × 24 = 6 gallons 
Concentration of water in the mixture = 100% – (concentration of sodium chloride) = 100% – 40% = 60% 
Initial amount of water in the tank = 60% × (total mixture)= 0.6 × 6 = 3.6 gallons 

Next, let’s find the amount and concentration of water after 2 hours: 
Amount of water that will evaporate in 2 hours = (rate of evaporation)(time) = 0.5(2) = 1 gallon
Remaining amount of water = initial amount – evaporated water = 3.6 – 1 = 2.6 gallons 
Remaining amount of mixture = initial amount – evaporated water = 6 – 1 = 5 gallons 
Concentration of water in the mixture in 2 hours =  

The correct answer is C.

One of the most effective ways to solve problems involving formulas is to pick numbers. Note that since we are not given actual values but are asked to compute only the relative change in the useful life, we can select easy numbers and plug them into the formula to compute the percentage increase. Let’s pick d = 3 and h = 2 to simplify our computations: 

Before the change: d = 3, h = 2; u = (8)(3)/22 = 24/4 = 6 
After the change: d = (2)(3)= 6, h =2/2 =1; u = (8)(6)/12 = 48 

Finally, percent increase is found by first calculating the change in value divided by the original value and then multiplying by 100:
(48 – 6)/6 = (42/6) = 7
(7)(100) = 700%

The correct answer is D.

Since there are variables in the answer choices, as well as in the question stem, one way to approach this problem is to pick numbers and test each answer choice. We know that x is m percent of 2y, so pick values for m and y, then solve for x. 
y = 100 
m = 40 
x is m percent of 2y, or x is 40 percent of 200, so x = (0.40)(200) = 80. 

So, for the numbers we are using, m is what percent of x? Well, m = 40, which is half of x = 80. Thus, m is 50 percent of x. The answer choice that equals 50 will be the correct choice. 

(A) y/200 = 100/200 = 0.5 WRONG 
(B) 2y = (2)(100) = 200 WRONG 
(C) 50y = (50)(100) = 5000 WRONG 
(D) 50/y = 50/100 = 0.5 WRONG 
(E) 5000/y = 5000/100 = 50 CORRECT 

Alternatively, we can pursue an algebraic solution. 

We are given the fact that x is m percent of 2y, or x = (m/100)(2y) = my/50. Since the question asks about m (“m is what percent of x?”), we should solve this equation for m to get m = (50/y)(x). 

Putting the question “m is what percent of x?” into equation form, with the word “Answer” as a placeholder, m = (Answer/100)(x). 

Now we have two equations for m. If we set them equal, we can solve for the “Answer.” 
(Answer/100)(x) = (50/y)(x) 
(Answer/100) = (50/y) 
Answer = 5000/y 

The correct answer is E.

The easiest way to solve this problem is to use the VIC method of substituting actual numbers for x and y. The problem asks us to take x% of y and increase it by x%. Since we are dealing with percentages, and the whole (y) is neither given to us nor asked of us, let's set y = 100 and x = 10. Note that this is a variation on the typical method of picking small numbers in VIC problems. 

10% of 100 is 10. Increasing that 10 by 10% gives us 10 + 1 = 11. Therefore 11 is our target number. Let's test each answer choice in turn to see which of them matches our target number.

(A) 100xy + x = 100(10)(100) + 10 which doesn't equal 11.

(B) xy + x/100 = 10(100) + 10/100 which doesn't equal 11.

(C) 100xy + x/100 = 100(10)(100) + 10/100 which doesn't equal 11.


which doesn't equal 11


The correct answer is E

First, determine the total cost of the item at each store. 

Store A: 
 $60 (MSRP) 

Store B: 
 $60 (MSRP) 

 

The difference in total cost, subtracting the Store B cost from the Store A cost, is thus $75.60 - $73.71 = $1.89. 

The correct answer is D.
 

Given an initial deposit of $1,000, we must figure out the ending balance to calculate the total percent change. 

After the first year, Sam's account has increased by $100 to $1,100. 

After the second year, Sam's account again increased by 10%, but we must take 10% of $1,100, or $110. Thus the ending balance is $1,210 ($1,100 + $110). 

To calculate the percent change, we first calculate the difference between the ending balance and the initial balance: $1,210 – $1,000 = $210. We divide this difference by the initial balance of $1,000 and we get $210/$1,000 = .21 = 21%. 

The correct answer is C.