Arithmetic (FDPR) Flashcards
Translate percent change from English to Math:
- 100% greater than original
- 45% less than original
- 130% of the original
- 75% of the original
- 200% of the original or 2x original
- 55% of the original
- 30% greater than the original
- 25% less than the original
What are the relative values on GMAT?
fractions, decimals, percents or ratios (proportion)
What are the concrete values on GMAT?
specific number of tickets sold, liters of water, etc
T or F: For positive numbers, if the starting faction is less than 1, the value fraction decreases - toward to 0- as you add the same number to the top and bottom
F
the fraction increase as it gets closer to 1
How does a faction value (that is greater than 1) change when you add the same number to top and bottom?
the result is smaller (getting closer to 1)
E.g: 3/2 > 4/3
List all strategies that you can approach for this problem
Which fraction is greater: 7/9 or 4/5?
1) The Double-Cross: 35/45 < 36/45
2) Percentage Exchange: 77.7% < 80%
What is the best strategy for this problem?
A TV originally priced at $700 was offered at a 12% discount. What was the sale price of the TV?
Benchmark Computation
- Amount of Reduction = 10% + 1% + 1% = 70 + 7 + 7 = 84
- Final amount = 700 - 84 = 700 - 80 - 4 = $616
Whats your approach?
65% of students have taken language classes. Of those students, 40% have studied more than one language. If there are 300 students, how many have studied more than one language?
(A) 78 (B)102. (C)120. (D) 150. (E)195
Look at the answer choices, then decide to do estimation or precise cal
- in this case, estimation is more efficient :
40% of 60% of 300 = 72 so ans must be greater but closed to choice (A)
When can you really use the estimation strategy?
1) When the problem includes “approximately” in its question
2) When the problem contains annoying/non-nice ratio number
3) When the value of answer choices vary greatly
i.e: (A) 780 (B) 2700 (C) 4320. (D) 6480. (E) 8400
When you see not nice % with decimals, what techniques to use?
When Leo imported a certain item, he paid a 7 percent import tax on the portion of the total value of the item in excess of $1,000. If the amount of the import tax that Leo paid was $87.50, what was the total value of the item?
1) look at the answer choices to see whether to use estimation
2) Apply Benchmark Computation:
- 7% of $1000 is $70 so the excess amount must be greater than 1,000
Simplify 60% of 45.35
3/5 x (45 +0.35) = 9x3 + 0.7x 3 = 27 +0.21= 27.21
How would only quarters and dimes add up to $2.50?
10 quarters, 0 dimes
8 quarters, 5 dimes
6 quarters, 10 dimes….
- The inverse proportion relationship is 2:5
For company A, by what percent did profit increase this over last?
1) Last year’s profit is 20% less than this year’s profit
2) Last year’s profit was 80k
Statement 1) or (A) is sufficient
- When we have the profit ratio (this/last) = 5/4, we can infer that there was 1/4 or 25% increase in profit
How do you solve the relative value of two departments within a company, when there are increase/decrease in revenue, profit, cost?
Weighted Average: Push-Pull in the increase/decrease between two apartments
What is the faster way to solve beside algebra?
A certain company that sells only cars and trucks reported that revenues from car sales in 1997 were down 11 percent from 1996 and revenues from truck sales in 1997 were up 7 percent from 1996. If total revenues from car sales and truck sales in 1997 were up 1 percent from 1996, what is the ratio of revenue from car sales in 1996 to revenue from truck sales in 1996?
Weighted Average
What technique you need to solve this efficiently?
If 65 percent of a certain firm’s employees are full-time and if there are 5,100 more full-time employees than part-time employees, how many employees does the firm have?
Ratio Difference and Multiplier X Ratio:
- F: P = 65% : 35%, then F-P = 30. X = 5,100
- X= 170 then Total employees = 170x100 = 17,000
What technique you need to solve this efficiently?
Karen bought a new television, originally priced at $690. However, she had a coupon that saved her $67. For what percent discount was Karen’s coupon?
Benchmark Estimation:
- 10% of $690 is 69 but since $67<69 so the discount must be smaller than 10%
what is your foremost step set-up?
If y ≠ 3 and 3x/y is a prime integer greater than 2, which of the following must be true?
a. x = y
b. y = 1
c. x and y are prime integers.
Just like those problems in data sufficiency, you set up test cases in which:
- 3x/y = 3, 5, 7, 11, 13,.. -> x/y = 1, 5/3, 7/3, 11/3… From the pattern, we can eliminate a & b
- Next, since its the ratio, always do extreme cases by doubling both numerator & denominator (i.e: x/y =10/6) -> we can eliminate c as well
What is the common theme in diguise?
A $500 investment and a $1,500 investment have a combined yearly return of 8.5 percent of the total of the two investments. If the $500 investment has a yearly return of 7 percent, what percent yearly return does the $1,500 investment have?
Weighted Average
For a recent play performance, the ticket prices were $25 per adult and $15 per child. A total of 500 tickets were sold for the performance. If the avg tix price is $21, how many of the tickets sold were for adults?
Weighted Average Ratio Line approach
What is the lesson/hidden theme for this question?
If y is greater than 110 percent of x, is y greater than 75 ? Is (2) sufficient?
(2) y - x =10
1) Distinguish btw “is greater than 110% of x” & “is 110% greater than x”
2) A better rephrase approach: Don’t plug in the variables into static equation
-1.1x -x = 10 -> x =100 (Yes) BUT
- 1.5x - x = 10 -> X = 20 (No: y is not greater than 75)
3) Always Extreme test cases for two unkown variables
Key Lesson
Was the number of books sold at Bookstore X last week greater than the number of books sold at Bookstore Y last week?
(1) Last week, more than 1,000 books were sold at Bookstore X on Saturday and fewer than 1,000 books were sold at Bookstore Y on Saturday
(2) Last week, less than 20 percent of the books sold at Bookstore X were sold on Saturday and more than 20 percent of the books sold at Bookstore Y were sold on Saturday
1) More units comprise of a smaller portion of a total indicate that the total (denominator) must be large
2) Fewer units comprise of a larger portion indicate total units must be small
What does this really test?
Tickets for all but 100 seats in a 10,000-seat stadium were sold. Of the tickets sold, 20% were sold at half price and the remaining tickets were sold at the full price of $2. What was the total revenue from ticket sales?
A. $15,840
B. $17,820
C. $18,000
D. $19,800
E. $21,780
**The Law of Large Number Estimation: **
Since 100 seats are comparatively small to 10,000 seats, we can take 20% of full 10,000, we then have:
- 2000x $1 + 8000x$2 = $18,000
- Now take into the account of 100 seats, our answer must be less than but close to $18,000
What the best approach for this problem?
During the four years that Mrs. Lopez owned her car, she found that her total car expenses were $18,000. Fuel and maintenance costs accounted for 1/3 of the total and depreciation accounted for 3/5 of the remainder. The cost of insurance was 3 times the cost of financing, and together these two costs accounted for 1/5 of the total. If the only other expenses were taxes and license fees, then the cost of financing was how much more or less than the cost of taxes and license fees?
1) Best to avoid computation for the concrete value from every fraction/ratio step
2) Combine/Subtract/ Multiply multitudes fractions so they can cancel out for the final fraction.
How do these two statement differ? (1) “Y is greater than 110 percent of x” vs (2) “Y is 110% greater than X”
(1) y > 1.1x (so y can be 1.15x, 1.2, 1.5x..)
(2) y = 1.1x
For the total # of tickets X, the revenue for adult tickets and children tickets is Y, what is the range of possibility for the average cost, given that adult ticket (cost A) > children ticket (cost B)?
The average cost will fall in between cost A & B - i.e: B < Y/X < A
Translate to math expression
After paying a 10 percent tax on all income over $3,000, a person had a net income of $12,000. What was the income before taxes?
$12,000 = $3,000 + (X-3000)x0.9
Try intuition method to solve
450 is what percent greater than 15 ?
450 is equal to 30 times 15 or 3000% of 15 - equivalent to:
- 29 times greater than 15
What’s your strategy?
Bottle R contains 250 capsules and costs $6.25. Bottle T contains 130 capsules and costs $2.99. What is the difference between the cost per capsule for bottle R and the cost per capsule for bottle T?
(A) $0.25 (B) $0.12 (C) $0.05 (D) $0.03 (E) $0.002
1) Look at the answer choice for the widespread range to decide estimation or precise cal
2) Make decimals into a nice number for easy calculation: $6.25 = 625 cents
3) Notice 130 caps costs $3, so $6 or 600 cents will cost around 260 caps -> Bottle R & T difference is 25cents/250 capsules ~ 0.1 cents or Answ E)
W/o precise calculation, estimate w/ the base rate
On July 1, 1982, Ms. Fox deposited $10,000 in a new account at the annual interest rate of 12 percent compounded monthly. If no additional deposits or withdrawals were made and if interest was credited on the last day of each month, what was the amount of money in the account on September 1, 1982?
(A) $10,200 (B) $10,201
1) Assuming 12% rate compound annually, we will have $1200/year or $100/month. Hence, 2 months will have $10,200
2) However, because the rate compound monthly, the precise number must be greater than $10,200
Practice the translation in term of profits, original price & sale price
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?
10% Profit represent Rev = 1.1C
- Discounted Sale Price (Rev) = 1.1 Manufacture Price (Cost)
-> 440 x4/5 = 1.1 C -> C = 320
Why you can’t just flip the ratio 3:4 -> 4:3 for the invested amount?
A total of $60,000 was invested for one year. Part of this amount earned simple annual interest at the rate of x percent per year, and the rest earned simple annual interest at the rate of y percent per year. If the total interest earned by the $60,000 for that year was $4,080, what is the value of x?
(1) x = 3y/4
The flip for the ratio invested amount is the result of the ratio of the distance difference from the mean:
- (Avg % - x)/(y - Avg %) = Unknown (given Avg % = 4,080/60,000)
- This is not the same as ratio x/y = 3/4
- In addition, x & y can have various values such as 6 & 8, or 9 &12 so these distance difference can vary
What is the key lesson for this problem?
Mr. Kramer, the losing candidate in a two-candidate election, received 942,568 votes, which was exactly 40 percent of all votes cast. Approximately what percent of the remaining votes would he need to have received in order to have won at least 50 percent of all the votes cast?
A. 10%
B. 12%
C. 15%
D. 17%
E. 20%
You are given a very nasty large number -> Don’t try to do pure math. Think critically:
- the # of kramer’s vote = k = 0.4 T (total votes) & he needs 0.1T to win the election
- the remaining votes = 0.6T
- 0.1T = 0.6T (x/100) –> X = 100/6 % or 1/6 = 17%
Identify a specific info in stimulus that makes (1) actually sufficient
Last year Luis invested x dollars for one year, half at 8% annual interest and the other half at 12% annual interest. Now he wants to reinvest the x dollars for one year in the same two types of investments, but the lower rate has decreased. If the higher rate is unchanged, what fraction of the x dollars must he reinvest at the 12% so that the total interest earned from the x dollars will be the same for both years ?
(1) The lower rate is now 6 percent.
Equal amount of x is invested in each account - 1:1 ratio, hence
Method 1: Draw the ratio line, we have the WA is 10%
- Since after the % change, the total interest is still the same, unchanged WA (10%) can be applied to find the new ratio for 6% &12% IRR
- From the new ratio, we can find relative value for X
**Method 2: **
- 8% . X/2 + 12%. X/2 = A. 12% + (X -A). 6%
- After simplification, we can find ratio A/X
Apply Ratio Technique to solve
A merchant purchased a jacket for $60 and then determined a selling price that equalled the purchase price of the jacket plus a markup that was 25 percent of the selling price. During a sale, the merchant discounted the selling price by 20 percent and sold the jacket. What was the merchant’s gross profit on this sale?
Revenue (R) = Cost (C) + 0.25R -> 0.75R = C or
- R : C: Profit (P) = 4:3:1
The revenue after 20% discount: 4/5R -> the new ratio:
- R : C : P = 4x4/5 : 3 : (16/5-3 = 1/5)
W/ C = $60 we know the new P = $20x 1/5 = $4
What is the most effect method?
The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers?
Multiplier X in Ratio:
- (30X + 50)/ (X +5) = 25/1
Key lesson
For each trip, a taxicab company charges $4.25 for the first mile and $2.65 for each additional mile or fraction thereof. If the total charge for a certain trip
was $62.55, how many miles at most was the trip?
DON’T ALWAYS fall for the hard-division for calculation: (62.55- 4.25)/ 2.65 = 58.3/2.65
- $2.65 x 10 = 26.50, 26.50 x 2 = $53.00 (20 miles)
- 58.30 - 53.00 = 5.30 =2 extra miles
- Total = 22 extra miles + First mile = 23
You got lucky w/ this problem - Wrong approach but right answer
A garden center sells a certain grass seed in 5-pound bags at $13.85 per bag, 10-pound bags at $20.43 per bag, and 25-pound bags at $32.25 per bag. If a customer is to buy at least 65 pounds of the grass seed, but no more than 80 pounds, what is the least possible cost of the grass seed that the customer will buy?
Key Lesson: This is hidden min/max problem. Be wary of your assumption:
- You can’t simply assume that 75 lbs gives the best price:
- Perhaps the combination of price right at the minimum quantity (65lbs) can be cheaper than those of 75lbs and still statisfy the given condition
- Cal the price for: 2x25lbs + 10lbs + 5lb = 98.70 > 96.75 (of 3x25lbs)
Interprete this problem in multiple ways
Stock A cost $120 and Stock B costs $20. What percent less than the price of Stock A is the price of stock B?
Use analogy: 10% less of 100 = 90% remaining of 100
- 20 is 1/6 remaining of 120, hence, 5/6 less than 120
- 20 = 120 - 120. (X/100) -> X = 100/120 = 5/6
Use WA technique, but w/ grams of protein (not %)
A rabbit on a controlled diet is fed daily 300 grams of a mixture of two foods, food X and food Y. Food X contains 10% protein and food Y contains 15% protein. If the rabbit’s diet provides exactly 38 grams of protein daily, how many grams of food X are in the mixture.
38g lies between the max protein of Food X & max protein of Food Y
- max of Food X = 10% of 300 = 30
- max of Food Y = 15% of 300 = 45
From this relationship we can find the ratio of X to Y
Why Weighted Average can backfire?
On Monday, a person mailed 8 packages weighing an average (arithmetic mean) of 12 (3/8) pounds, and on Tuesday, 4 packages weighing an average of 15 (1/4) pounds. What was the average weight, in pounds, of all the packages the person mailed on both days?
1) The fraction of the ratio calculation are not nice
2) Critical thinking to use conventional strategy to cancel out the denominators
Which target smart-Number do you use to avoid complex fraction?
Mary’s income is 60 percent more than Tim’s income, and Tim’s income is 40 percent less than Juan’s income. What percent of Juan’s income is Mary’s income?
(A) 124% (B) 120% (C) 96% (D) 80% (E) 64%
We find the restriction is at Tim’s income = 3/5 Juan, so we pick smart number for Juan = 150
% k increase vs % k change
What do you to have be cautious about when Pick-Number & Back-solving?
You have to account for the sign when you back-solving with the pick-number (i.e: k=10):
- 10% increase mean k = 10 when plug-in
- 10% change mean k = +/- 10
Key lesson:
If 3/7 of the students in a room are seniors and 7/25 of the other students are juniors, and there are x students in the room who are not juniors or seniors, how many students are in the room?
NOT EVERY QUESTION should be treated as Double Matrix
- Even if you tried DM method, focus on re-thinking your assumption & switch different strategy
What is the average rate of changes?
Its the slope of the linear equation of two points (a,b) & (x,y) - Rise/Run
Practice w/ Everyday Math Mentality
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 ?
Since both Kevin’s and Jean’s salaries increase by the same percentage, the difference between their salaries will likewise increase by that percentage. Assuming an initial difference of $100 in 1998, it would rise to $1001.03 = $103 in 1999 (a 3% increase) and further to $1031.03 = $106.09 in 2000 (an additional 3% increase). This results in a total increase of 6.09%.
Key Lesson?
At a recent sale, Harue bought various numbers of 5 types of school supplies - markers, pens, pencils, erasers, and notebooks - each advertised with a different pricing strategy. Harue bought 4 markers for the price of 3, 3 pens for the price of 2, 5 pencils for the price of 3, 2 erasers for the price of 1, and 5 notebooks for the price of 2. One of the 5 types of school supplies was advertised as “Buy one, get one at 50% off.” Which type?
1) Don’t get distracted by the words “different pricing” - We can pick number to do small test cases instead of variables
2) Cost = $1 then “buy one, get one at 50%” will cost $1.5 instead of $2
- Since variables are units & not the price, we have the total cost in ratio - 2:1.5 or 4:3 or 8:6
Why you can’t just flip the ratio 3:2 -> 2:3 for the invested amount?
A total of $60,000 was invested for one year. Part of this amount earned simple annual interest at the rate of x percent per year, and the rest earned simple annual interest at the rate of y percent per year. If the total interest earned by the $60,000 for that year was $4,080, what is the value of x?
(2) The ratio of the amount that earned interest at the rate of x percent per year to the amount that earned interest at the rate of y percent per year was 3 to 2.
The flip for the ratio invested amount is the result of the ratio of the following:
- (4080-A)/ (B - 4080) = 2:3 (given A & B are the amount that earned 100% on either x% or y% interest, respectively)
If rate 1produces 20 toys per hour & rate 2 produces 30 toys per hour, what is the new rate if 40% of rate 1 machine replaced w/ rate 2 machine?
Just as the variables are number of machines & not the rate, we have the new rate in according to the proportion of each type of machine:
- 3/5(20) + 2/5(30) = 12 + 12 = 24 toys per hour
What is the TRAP? What does GMAT subtly testing?
Half of a large pizza is cut into 4 equal-sized pieces, and the other half is cut into 6 equal-sized pieces. If a person were to eat 1 of the larger pieces and 2 of the smaller pieces, what fraction of the pizza would remain uneaten?
1) Even though each type of pieces are pizza, they don’t have the common units (not the same size) to compare to total
2) Divisibility - Pick 24 = total # of common units (C) :
- Eat 1 of 4 equal-sized of 12 C = 3C
- Eat 2 of 6 equal-sized pieces of 12C = 4C
- Uneaten = 24-7 = 17C -> Ratio = 17/24
- 10% greater than original is __
- 75% of the original __
110% of the original
25% less than the original
How do you convert 17/25 to percentage?
Multiply by 4 for numerator and denominator. We have 68/100 = 68%
11.1 % equal to fraction of
12.5% equal to fraction of
16.7% equal to fraction of
175% equal to fraction of
1/9
1/8
1/6
7/4
83.3% equal to fraction of
125% equal to fraction of
133% equal to fraction of
5/6
5/4
4/3
The cool trick: Divide by 5
1) Double the dividend
2) Move decimal one place to the left
i.e: 8/5 = 1.6
435/5 = 87.0
The cool trick: Multiplying with the decimals ending in 5
Multiplying by halving and doubling:
1) 16x 5.5 = 8 x11 = 88
2) 2.25x 36 =4.5x18 = 9x9 = 81
What is the devil trap?
If a certain toy store’s revenue in November was 2/5 of its revenue in December and its revenue in January was 1/4 of its revenue in November, then the store’s revenue in December was how many times the average (arithmetic mean) of its revenues in November and January?
After you lay out the ratio map, you have to convert all variables to matching unit of multiple x:
- N : D = 2: 5 & N: J = 4:1 => N:D:J = 4:10:1, hence
- D: Avg (N &J ) = 10/ [(4+1)/2] = 4:1
NOTE: NOT - 5/ [(4+1)/2]
