Quant practice (error log) Flashcards
(10⁵ − 2)(10⁵ + 2)
=(10⁵ − 2)² − 4
This isa² − b²= (a − b)(a + b). Don’t expand 99,999² — it’s a trap.
Evaluate the squares using algebraic identity
Alfredo invested a total of $33,000 in 2 accounts, A and B, with annual interst rates of 5% and 3% respectively. for the first year the interest earned bu Account A was twice as much the interest earned by Account B. What was the interest earned by the 2 accounts in the first year ?
- Let’s pretend B earns$1in interest.
- Then A earns$2, right?
- How much money would need to be in B to earn $1 at 3%? → $1 ÷ 0.03 = ~$33.33
- How much in A to earn $2 at 5%? → $2 ÷ 0.05 = $40
Now, total investment = ~$33.33 + $40 = ~$73.33
So, if $73.33 gives $3 interest, then $33,000 will give:
Proportional:33,000/ 73.33 ≈450×3=1350
Yuriko drove from Ashland to Bayville at an average rate of 40 miles per hour and from Bayville to Cabot at an average rate of 50 miles per hour. If Yuriko’s trip from Ashland to Bayville was 90 miles longer and took 3 hours more than her trip from Bayville to Cabot, how many hours did her trip from Ashland to Bayville take?
We need:
A distance for the first trip that is 90 miles longer than the second
And a time that is 3 hours longer
- Distance A→B = 6 × 40 = 240 miles
- B→C is 90 miles shorter = 150 miles
- Time for B→C = 150 ÷ 50 = 3 hours
- Time difference = 6 − 3 = 3 hours ✅
Boom. No need for equations.
✅ Answer:6 hours
Distance = Rate × Time (D = RT)
A certain economic indicator is proportional to the square of the unemployment rate. If the unemployment rate increases by 10 percent, then the indicator will increase by
Direct Proportion: If Indicator ∝ (Unemployment Rate)², then:
NewIndicator = (NewRate)²
Let original unemployment rate = 10
Then the original indicator = 10² = 100
New unemployment rate (10% higher) = 11
New indicator = 11² = 121 121-100= 21 → 21%
If a = √2- 1 and b = 1, which of the following is equal to [a² + b(b + 2a)] + [b²=+ a(a + 2b)] ?
Algebraic expressions,simplifying and rules square roots
2a² + 2b² + 4ab
= 2( a² + b² + 2ab)
= 2 (a + b)²
= 2 (√2- 1 + 1)²
= 2 (√2) ²
= 2x2=4
Expand both groups, simplify, factor into 2(a+b)² , then plug in.
A certain identification code consists of 6 digits, each of which is chosen from 1, 2, …, 9. The same digit can appear more than once. If the sum of the 1st and 3rd digits is the 5th digit, and the sum of the 2nd and 4th digits is the 6th digit, how many identification codes are possible?
Combinatorics
You need to count how many 6-digit codes you can create using the digits 1 through 9 (repeats allowed), under two conditions:
Digit 5 = Digit 1 + Digit 3
Digit 6 = Digit 2 + Digit 4
Digits allowed = 1 to 9
The smallest sum = 1 + 1 = 2
The largest sum = 9 + 9 = 18
Total valid pairs for sums 2 through 9:
1+2+3+4+5+6+7+8= 36
There are 36 valid ways to choose (Digit 1, Digit 3, Digit 5)
and 36 valid ways to choose (Digit 2, Digit 4 and Digit 6)
so = 36x36= 1296
The average age of 3 brothers is 18 years. Their ages are in the ratio 2:3:4. What is the age, in years, of the youngest brother?
Ratio problems & averages
2 + 3 + 4 = 9 parts total of all their ages is:
3 × 18 = 54 years
54÷9=6Now multiply each brother’s “part” by 6:
Youngest: 2 × 6 = 12
Middle: 3 × 6 = 18
Oldest: 4 × 6 = 24
In 1998 Jean’s annual salary was greater than Kevin’s annual salary. In each of 1999 and 2000, Jean’s annual salary was 3 percent greater in that year than in the preceding year and Kevin’s annual salary was also 3 percent greater in that year than in the preceding year. By what percent did the difference between Jean’s and Kevin’s annual salaries increase from 1998 to 2000?
Four staff members at a certain company worked on a project. The amounts of time that the four staff members worked on the project were in the ratio 2 to 3 to 5 to 6. If one of the four staff members worked on the project for 30 hours, which of the following CANNOT be the total number of hours that the four staff members worked on the project?
Ratio problem
2x + 3x + 5x + 6x = 16x
then
if 2x = 30, x= 15, and 16x = 240
if 3x = 30, x=10 and 16x= 160
if 5x = 30, x= 6 and 16x = 96
if 6x = 30, x=5 and 16x = 80
so the answer is whatever option that is not those.
If ½ of the money in a certain trust fund was invested in stocks, 1/4 in bonds, 1/5 in a mutual fund, and the remaining $10,000 in a government certificate, what was the total amount of the trust fund?
Operations with fractions and whole numbers
The sum of the fractions is 19/20, which means 10,000 is 1/20 of the money. 10,000x 20 = 200,000
Each book in a certain library is classified in one of three categories: fiction, nonfiction, or biography. The ratio of the number of fiction books to the number of nonfiction books is 4 to 3, and the ratio of the number of nonfiction books to the number of biographies is 3 to 2. If there are at least 10,000 books in the library, what is the least number of books that could be in the library?
Chained ratio problem
An art auction house makes a commission of 5 percent on the first $10,000 of the selling price of a painting, 3 percent on the next $8,000, and a certain percent on any amount thereafter. If the auction house made a commission of $800 on a painting that sold for $22,000, what was the percent of the commission on the amount in excess of $18,000 ?
A furniture dealer bought a sofa at the manufacturer’s price and sold it at a 20 percent discount off its regular retail price of $440. If the dealer made a 10 percent profit on the manufacturer’s price of the sofa, what was the manufacturer’s price?
A contractor hired two paving firms to pave a parking lot. The first firm was to pave 5/8 of the parking lot, and the second firm was to pave the rest. On the first day, the first firm paved 2/3 of its portion and the second paved 1/2 of its portion. What fraction of the parking lot was NOT paved on the first day?
Find average
The average (arithmetic mean) selling price of 5 houses in a certain neighborhood was $250,000. If the average selling price of 3 of the houses was $280,000, what was the average selling price of the other 2 houses?
Arithmetic statistics
250x5 = 1250
and 380x3 = 840
so 1250-840 = 410
then 410/2 = 205
Four employees, Mark, Ellen, Dennis, and Jane, were given a certain amount of money to spend on lunch. Mark spent $3 less than of 3 less than the 1/3 of the total amount; Ellen spent $2 more than 1/4 of the amount; and Dennis spent 1/4 of the total amount; and Jane spent $7. If all of the money, and no additional amount, was spent, what was the total amount given to the four employees?
George’s home is 20 miles from his place of work. One day as he is driving home from work, his car breaks down when he is 5 miles from work. He starts walking toward his home at an average rate of 4 miles per hour. If d represents his distance from home, in miles, and t represents the time, in hours, that he has been walking, which of the following gives d in terms of t?
At a certain photoprocessing shop, the first standard-size print of a negative costs $4, and each additional print of the same negative costs $1. What is the total cost, in dollars, of y standard-size prints of each of x different negatives?
On a 600‐kilometer trip, a car traveled half the distance at an average speed of 60 kilometers per hour (kph) and the other half at an average speed of 100 kph. The car did not stop between the two halves of the trip. What was the car’s average speed during the trip as a whole?
600/2 = 300
time = d/r
first 300/60 = 5
second 300/100 = 3
total time = 8 h,
then average speed = d/t = 600/8 = 75
.Dan was driving when he and Alice left their office in a company car to travel to a business meeting. When Dan had driven of the distance from their office to the meeting place, his cell phone rang. He pulled off the road to answer the call, and Alice drove the rest of the way to the meeting place. If Alice drove 18 fewer miles than Dan, how many miles was their office from the meeting place?
Exactly 18 months ago management in a certain retail electronics store began making monthly observations of the percentage of shoppers who appear to be showrooming—examining a product while in the store and then buying the product online from another store. The results of the observations were that the percentage of customers who appeared to be showrooming increased by 0.5% each month. If p is the percent of customers who appeared to be showrooming in their store x months after the monthly observations began and 10 months ago p was equal to 10.5%, which of the following equations most accurately models the findings of the management team for the past 18 months?
If 3p = 4s, 5s = 3/r and r is not 0 , what is r in terms of p?
find average
A student’s average (arithmetic mean) test score on 4 tests is 78. What must be the student’s score on a 5th test for the student’s average score on the 5 tests to be 80 ?
Arithmetic statistics
4(78) = 312
5(8)= 400
then 400-312= 88
In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was three fifths After 600 additional Republicans and 500 additional Democrats registered, the ratio was four fifths. After these registrations, there were how many more voters in the district registered as Democrats than as Republicans?
Let R and D be the numbers of registered Republicans and registered Democrats, respectively, before the additional registrations.
Then, it is given that R over D = three fifths or 5R = 3D.
Also, 5(R + 600) = 4(D + 500)
5R + 3,000 = 4D + 2,000
3D + 3,000 = 4D + 2,000
since 5R = 3D from above
1,000 = D
Since D = 1,000
and 5R = 3D,
it follows that R = 600.
Finally, after the additional registrations, the number of Democrats was 1,000 + 500 = 1,500
and the number of Republicans was 600 + 600 = 1,200.
The difference is 1,500 − 1,200 = 300.
