Combined Formatting for QA Flashcards
QA tests
There are 5 women and 4 men. How many different teams of 2 women and 2 men are possible?
Count the women pairs and men pairs separately, then multiply for the total.
Women: 5 * 4 / 2 = 10
Men: 4 * 3 / 2 = 6
10 * 6 = 60
There are 9 candidates for 4 positions on a committee. How many different committees are possible?
126
out of 9, pick 4
4 slots on top, then divide by 4!, because the order doesn’t matter.
6 athletes are competing for gold, silver and bronze medals, plus 1 honorable mention. How many different ways can the awards be distributed?
This time, the order matters, so we don’t divide by “4!”
4 Slots (for gold, silver, bronze, honorable mention) –> 6*5*4*3 = 30*12 = 360
What is the sum of the odd integers from 5 to 25, inclusive?
“inclusive” means including 5 and 25.
Sum = Number of values (N) * Average Value
= N * (First + Last) / 2
N = (Last - First) / Increment + 1
= (25 - 5) / 2 + 1 = 20/2 + 1 = 11
Sum = 11 * (5+25) / 2 = 11 * 15 = 10*15 + 15 = 165
Mary spent a total of $10,000 on supplies, taxes, and rent last year. How much did she spend on rent?
(1) The amount spent on rent was 50 percent of the amount spent on taxes and supplies.
(2) The amount spent on supplies was 1/3 of the amount spent on taxes and rent
A- (1) Alone
DON’T CALCULATE!
Rephrase: 10,000 = S + T + R, R = ?
If we have 2 more unique equations, or 1 equation that has the combo “S + T”, it’s sufficient!
(1) R = .5(T + S) –> 2R = S + T –> Can substitute S + T into the equation, and solve for R.
(2) S = 1/3 (T + R). This time, we can’t isolate “S+T”. Insufficient.
If the product of the two digits of the positive integer N is 12, what is N?
(1) N > 40
(2) N is an odd integer
B: (2) Alone
Rephrase: what are the possibilities for N?
4*3 = 12, 2*6 = 12
So, N could be 43, 34, 26, or 62
1) Could be 43 or 62 –> Insufficient
2) Can only be 43 –> Sufficient
The number of pens that Joe and Mary had were in a 2:3 ratio. After Mary gave Joe 6 pens, Mary had 3 pens for every 7 that Joe had. After this gift, Joe had how many more pens than Mary?
8
Start with unknown multiplier: Joe / Mary = 2x / 3x
Then, we add 6 to Joe, and subtract 6 from Mary, to create an equation with the new ratio.
Make sure to put the ratio in the correct order: The new ratio is 7/3
Cross-multiply and solve for x.
Make sure to answer the question asked!
Plug in x –> (2x + 6) / (3x - 6) = 14 / 6 –> 14 - 6 = 8
A boat was sold at a loss of 40% of the purchase price. Another boat was sold at a profit of 20% of the purchase price. If each boat was sold for $30,000, what was the total profit or loss combined?
Let C = Cost ; calculate separately
Loss: 30,000 = (100% - 40%) * C
30,000 = .6C = 3/5 C
Divide by 3/5 –> Multiply by 5/3 –> C = 50,000 –> Loss = 20,000
Profit: 30,000 = (100% + 20%) * C = 1.2C = 6/5 C
C = 25,000 –> Profit = 5000
Combined, 20,000 - 5,000 = $15,000 Loss
Find the ratio of boxes loaded– night crew : day crew
3/4 the speed, 4/5 as many workers
3/4 * 4/5 = 3/5 = night crew : day crew.
Read carefully! Question asks what fraction of the TOTAL is the day crew
5 / (3+5) = 5/8
Shirts were sold for $20 more than the cost. They were sold at a profit of 40% of the purchase price. The total revenue from shirt sales was $2800. How many shirts were sold?
40
C = Cost or Purchase Price S = Sales Price
20 = 40% * C
20 = 2/5 * C
C = 50
S = 50 + 20 = 70
Revenue = Sales Price * Number sold –> R = SN –> N = R/S
2800 / 70 = 40
If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 2^8, what is the least possible value of n?
10
“The product of all integers from 1 to n” translates to n! (n factorial)
We need to find the lowest factorial that includes 8 total 2’s as factors.
What is the area of the yellow shaded region?
What strategy can we use, if an area is hard to calculate directly?
Calculate the area of the entire region, then subtract the unshaded part.
Area of square = 16
For circle, D = 4, so r = 2
Area of circle = πr2 = π(22) = 4π
16 - 4π
Machine type A does a job in 8 hours, while machine type B does a job in 12 hours. If 2 machines of type A are used, how many machines of type B are needed to complete the job in 90 minutes?
Calculate the Work separately for A and B:
A: N = 2, R = 1/8, T = 90 minutes = 3/2 hours
W = NRT = 2 * 1/8 * 3/2 = 3/8
So, work remaining for type B is 1 - (3/8) = 5/8
B: 5/8 = N * 1/12 * 3/2
5/8 = N * 1/8 –> N = 5
DS: What is the remainder when the positive integer x is divided by 4?
(How can we translate the following statements? What is the pattern?)
(1) When x is divided by 8, the remainder is 1
(2) When x is divided by 6, the remainder is 5
A –> (1) Alone
(1) –> 1 more than a multiple of 8
1, 9, 17, 25, 33….
All are also 1 more than a multiple of 4 –> Sufficient
(2) 5 more than a multiple of 6 –> 5, 11, 17, 24…
5/4 – > remainder 1
11/4 —> remainder 3
Insufficient.
DS: What is the remainder when the positive integer x is divided by 8?
(How can we translate the following statements? What is the pattern?)
(1) When x is divided by 8, the remainder is less than 5
(2) When x is divided by 4, the remainder is 2
C: Together
(1) Clearly insufficient alone: R (remainder) can be 1, 2, 3, or 4
(2) 2 more than a multiple of 4
Pattern is 2, 6, 10, 14,
Remainders when above numbers are divided by 8: 2, 6, 2, 6
2 possibilities –> Insufficient
Together, statement (1) eliminates 6, so the reminder must be 2 –> Sufficient
Is b > 4 ?
(1) The point (3, b) lies outside the circle whose equation is x2 + y2 = 25
(2) The point (2, b) lies on the line whose equation is 2y - 8x = -4
B –> (2) Alone
(1) “lies outside the circle” –> x2 + y2 > 25
Plug in 3 for x: 9 + y2 > 25
y2 > 16
Trap: y > 4 OR y < -4 –> Insufficient
(2) “The point lies on the line” means the point satisfies the equation:
Plug in 2 for x: 2y - 16 = -4
2y = 12 –> y = 6 –> b = 6 –> Sufficient
There were 3 plums and 4 lemons. After more of each were added, the ratio of plums to lemons was 2:1. How many lemons were added?
(1) A total of 8 plums and lemons were added
(2) The number of plums added is 5 more than twice the number of lemons
A: (1) Alone
Need 1 more unique equation to solve.
(1) P + L = 8 Unique equation –> Sufficient
(2) Trap! Same as the info in the prompt, not a unique equation!
Cross-multiply the prompt equation to check if it’s the same as (2)
P = 5 + 2L
A rectangular garden has a perimeter of 140 feet. The ratio of the length to the width is 4:3. What is the area?
1200 ft2
P = 140 = 2L + 2W
70 = L + W
L : W = 4:3
May be able to quickly see, L = 40 , W = 30
Using equations, L = 4/3 W
Substitute in for L: 70 = 7/3 W –> W = 30 –> L = 40
A = WL = 30*40 = 1200 ft2
An integer between 1 and 66 is chosen at random. What is the probability that n(n+1) will be divisible by 6?
Pattern will repeat every 6 numbers, up to 66.
“Divisible by 6” means “has both 2 and 3 as factors”
Easiest way is to create a chart:
4/6 = 2/3
N is 3 times the average of 15 numbers. In terms of N, what is the average of all the numbers, including N?
Let A = average of the 15 numbers
N = 3A –> A = 1/3 N
It asks for the average of all 16 numbers : (N + 15A) / 16
Substitute for A –> N + 5N / 16
6N / 16 = 3/8 N
Mary drives at a speed of 66 feet/second. How many miles away is she from her home, if it will take her 3 hours to reach home? (Given 1 mile = 5280 feet)
A) 25
B) 80
C) 135
D) 210
E) 300
c) 135
RT = D ; T = 3 –> 3R = D
Need to convert from feet/second to miles/hour. Set it up so units cross out on top and bottom, to avoid mistakes. (see image below)
If we notice the answer chioces are quite far apart, we can approximate, instead of calculating exactly!
3600/5280 is approximately 2/3 –> 2/3 * 66 = 44 mi/hr. (3600/5280 simplifies to 15/22 –> you can verify this as practice, but if we can approximate and save time that’s great)
3R = D –> 44*3 = 132 –> C
If there are 3 green marbles, 2 yellow marbles, and 5 red marbles.
If 2 are selected without replacement,
A) what is the probability they are both red?
B) What is the probability of 1 green, 1 yellow?
There are 2 events, so we multiply the probabilities for picking each marble.
There are 10 total marbles to start, and 9 after picking one, so these are our denominators.
A) 1st red 5/10 , 2nd red 4/9 –> 5/10 * 4/9 = 1/2 * 4/9 = 2/9
B) This time, there are 2 options:
1st green 3/10, 2nd yellow 2/9 –> 3/10 * 2/9 = 1/15
1st yellow 2/10, 2nd green 3/9 –> 2/10 * 3/9 = 1/15
Total is 1/15 + 1/15 = 2/15
Is the probability greater than 50% that a student randomly selected will be a female with brown hair?
(1) 1/2 of the students are female
(2) 1/2 of the students have brown hair
D - Each Alone
If the question asked for the probability VALUE, we couldn’t find it.
But since it asks for YES / NO, we can definitively say NO for each statement.
The BOTH box on matrix (or green “Intersection” on Venn Diagram) can’t be greater than the total female or total brown hair!
Is the probability less than 40% that a student randomly selected will be a female with brown hair?
(1) 1/2 of the students are female
(2) 1/3 of the students have brown hair
B - (2) Alone
Could convert fractions to percents for easier comparison: 1/2 = 50%, 1/3 is about 33%.
(1) Maximum for Both is 50%. Could be YES or NO. Insufficient
(2) The maximum for Both is 1/3 –> Must be YES –> Sufficient.
The Both box on matrix can’t be greater than the Total for brown hair!
Joe passed a store while traveling east at a constant speed of 30 miles per hour at 3pm. 30 minutes later, Kate passed the same store while traveling east at a constant speed of 36 miles per hour. At what time would Kate catch up to Joe?
How far ahead is Joe? 30 minutes ahead –> convert to a distance
Convert 30 minutes to 1/2 hour
30 mph * 1/2 = 15 miles ahead
We can use relative rates: since they are heading the same direction, we find the difference:
Kate is catching up at a rate of 36-30 = 6mph
T = D/R = 15 miles / 6mph = 2.5 hours (from the time KATE passes the store)
Be careful not to miss a step: Joe passes the store at 3pm, so Kate passes it at 3:30.
3:30 + 2.5 hours = 6pm
Of 400 athletes, 160 neither ran nor jumped, and 120 jumped. For every athlete that did both, 3 only ran. What percentage of athletes both ran and jumped?
We want “Both”. So define x as “Both”.
Then, 3x goes in “only ran”.
400 - 120 = 280 in “Not J” box
Solve for x: 3x + 160 = 280 –> 3x = 120 –> x = 40
40 / 400 = 10%
Real estate expenses accounted for 40% of operating expenses. There were $8 million of rent expenses, or 1/4 of the real estate expenses. How many million dollars were operating expenses?
8 = 1/4 Real Estate Expenses (REE)
REE = 32
32 = .4 Operating Expenses (OE)
Easiest if you recognize .4 as 2/5 –> multiply by 5/2 on both sides.
32*5/2 = 80 million dollars
In the sequence 3, 9, 27, 81, … each term after the first is three times the previous term. What is the 10th term minus the 9th term?
A) 39
B) 2 * 39
C) 38
D) 2 * 38
B) 2 * 39
1st term is 31.
Following the pattern, 10th term is 310, 9th term is 39
So, find 310 - 39
Factor out 39 –> 39 (3 - 1)
2 * 39
If 4*2x + 2x+4 = 10*22y, what is y in terms of x?
Product Rule of Exponents: 2x+4 = 2x * 24
Factor out 2x –> 4*2x + 2x * 24 = 2x (4 + 24) = 20 * 2x
Divide by 10 on both sides: 20 * 2x = 10*22y —> 2*2x = 22y
Use product rule again: 2*2x = 22y –> 2x+1 = 22y
x + 1 = 2y
y = (x+1) / 2
3/4 of the men is equal to 1/6 of the total number of adults.
What is the ratio of women to men?
3/4 M = 1/6 A
Multiply by 4/3 on both sides –> M = (4/3) * (1/6) A = 2/9 A
If M = 2/9 A, then women are all the rest –> W = 7/9 A.
Ratio of women to men is 7:2
How many positive integers between 50 and 150 are multiples of 3 and 4?
Must be a multiple of 12.
(Last - First) / Increment + 1
First = 60
Last = 144
(144 - 60) / 12 + 1 = 84 / 12 + 1 = 7 + 1 = 8
Solve | 3x - 2 | = 4
Need to split into 2 equations:
A) 3x - 2 = 4
3x = 6 –> x = 2
B) If it’s negative, it flips sign.
Make sure to put negative outside of parenthesis
- (3x - 2) = 4 –> -3x + 2 = 4
- 3x = 2
x = -2/3
There were twice as many pets as adults and 4 times as many women as men. If there was a total of 60 men, women, and pets, how many women were there?
“twice as many pets as adults” –> 1/3 were adults –> 20 adults
4 times as many women as men –> 4/5 women
4/5 * 20 = 16
If n is a positive integer, what is the tens digit of n ?
(1) The hundreds digit of 10n is 5.
(2) The tens digit of n + 1 is 5.
A - (1) Alone
(1) If we multiply n by 10, the tens digit becomes the hundreds digit.
So, the tens digit of n is 5 –> Sufficient
(2) If n=50, then the tens digit of n is 5.
If n=49, then the tens digit of n is 4.
Insufficient
Working alone, pump X pumped out ¼ of the water in a tank in 2 hours.
Then pumps Y and Z started working and the three pumps together pumped out the rest of the water in 3 hours.
If pump Y, working alone, would have taken 18 hours to pump out the rest of the water, how many hours would it have taken pump Z, working alone, to pump out all of the water?
R = W / T
Rx = (1/4) / 2 = 1/8
Combined: “rest of tank” –> 3/4 of tank in 3 hours
Rx + Ry + Rz = (3/4) / 3 = 1/4
3rd sentence says Ry = (3/4) / 18 = 1/24
Rz = 1/4 - Rx - Ry
Find common denominator of 24
Rz = 6/24 - 3/24 - 1/24 = 2/24 = 1/12
12 hours
Joe bought a bike for $120 and then sold it for a 75 percent profit. If Joe sold it at a 1/8 discount off the original retail price, approximately what was Joe’s profit as percentage of the original retail price?
A) 22%
B) 28%
C) 33%
D) 38%
E) 45%
D: 38%
75 percent profit * 120 = 90
90 + 120 = 210 Selling price
Sold at a 1/8 discount –> 7/8 Retail price = 210
Retail price = 8/7 * 210 = 240
90 / 240 = 3/8 = 37.5% –> closest to 38%
