OG Problems Review

228 Quant Review DS small square touching sides of large square What is side length of larger square? 1) the side length of the smaller square is 10 2) Each vertex of the small square cuts 1 side of the larger square into 2 segment s with lengths in ratio of 1:2

1) NS - can't get larger side 2) larger square sides have a total of x+2x but no values so NS Both: value and ratios are enough along with Pythagorean theorem 10 = square root (x^2 + 2x^2) x can be solved to get the side of the larger square

For a party, three solid cheese balls with diameters of 2 inches, 4 inches, and 6 inches, respectively, were combined to form a single cheese ball. What was the approximate diameter, in inches, of the new cheese ball? (The volume of a sphere is 4/3 (Pi)(r^3), where r is the radius.)

1) figure out volume of all cheese balls r = 1,2,3 V=4/3*pi*(1^3 + 2^3 + 3^3) V=4/3*pi*(36) 2) Determine r for new ball 4/3*pi*(36) = 4/3*pi*(r^3) r = cube root of 36

Quant review 232 DS Is zw positive? 1) z+w^3 = 20 2) z is positive

1) if z=1, w = cube root of 19 and zw >0 if z = 20, w = 0 then zw is neither Remember 0 is possible and 0 is neither positive or negative NS 2) can use the same case as above - 2 different answers (yes and no) Both: same examples can be used and not sufficient E

Three-fourths of the area of a rectangular lawn 30 feet wide by 40 feet long is to be enclosed by a rectangular fence. If the enclosure has full width and reduced length rather than full length and reduced width, how much less fence will be needed?

3/4 (30)(40) = 900 30x = 900 x=30 Fence needed = 2(30+30) = 120 40x = 900 x=22.5 Fence needed = 2(22.5+40) = 125 Difference = 125-120 = 5

Sam has $800 in his account. He will deposit $1 in his account one week from now, $2 two weeks from now, and each week thereafter he will deposit an amount that is $1 greater than the amount that he deposited one week before. If there are no other transactions, how much money will Sam have in his account 50 weeks from now?

800 + 1 +2 ..49+50 800 + 25(51) 800+$1,275. Thus, after 50 weeks Sam will have $800 + $1,275 = $2,075.

The product of two negative numbers is 160. If the lesser of the two numbers is 4 less than twice the greater, what is the greater number?

Assign variables, create 2 equations - solve for greater number Let x and y be the two numbers, where x is the lesser of the two numbers and y is the number desired. From the given information it follows that xy = 160 and x = 2y - 4, from which it follows that (2y - 4)y = 160. Dividing both sides of the last equation by 2 gives (y - 2)y = 80. Thus, 80 is to be written as a product of two negative numbers, one that is 2 less than the other. Trying simple factorizations of 80 quickly leads to the value of y: (-40)(-2) = 80, (-20)(-4) = 80, (-10)(-8) = 80. Therefore, y = -8. Note that because -8 is one of the answer choices, it is not necessary to ensure there are no other negative solutions to the equation (y - 2)y = 80. Alternatively, (y - 2)y = 80 can be written as y^2 - 2y - 80 = 0. Factoring the left side gives (y + 8)(y - 10) = 0, and y = -8 is the only negative solution.

Each car is either blue or white. What is the average sticker price of cars? 1) 1/3 of all cars are blue and average 21000 2) 2/3 of all cars are white and average 24000

Avg = sum / n Sum = sum of blue cars prices + sum of white cars prices Avg = sum of blue/n + sum of white / n 1) Avg*n = sum of blue 21000*(1/3)n = 7000n Avg = 7000n/n + sum of white/n Avg = 7000 + sum of white/n Not sufficient 1) Avg*n = sum of white 24000*(2/3)n = 16000n Avg = Sum of blue/n + 16000n/n Avg = Sum of blue/n + 16000 Not sufficient Both: Avg = 7000 + 16000 Sufficient

If S= 1 + (1/(2^2)) + (1/(3^2))+ (1/(4^2)) +(1/(5^2))+(1/(6^2))+(1/(7^2))+(1/(8^2))+(1/(9^2))+(1/(10^2)), which is true? S > 3 S = 3 2 < S < 3 S = 2 S < 2

By appropriately grouping the fractions being added and then replacing each fraction within a group by the greatest fraction in that group, the value of S can be shown to be less than 2 without extensive calculation. S = 1 + (1/(2^2))+ (1/(2^2))+(1/(4^2))+(1/(4^2))+(1/(4^2))+(1/(4^2))+(1/(8^2))+(1/(8^2))+(1/(8^2))

Problem Solving 52 (OG) A flat patio was built alongside a house as shown in the figure above. If all angles shown are right angles, what is the area of the patio in square feet?

Calculate correctly! 15*40 + (35-15)*20 =600+(20*20) = 600+400 = 1000

At a certain fruit stand, the price of each apple is 40 cents and the price of each orange is 60 cents. Mary selects a total of 10 apples and oranges from the fruit stand, and the average (arithmetic mean) price of the 10 pieces of fruit is 56 cents. How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is 52 cents?

Can round it to be whole numbers to make calcs easier x = apples 10-x = oranges (total is 10) [40x+60(10-x)]/(10) = 56 (solve for x) x = 2 (oranges = 8) Second scenario (plug in apples and oranges are 8-y) [40(2) +60(8-y)] / (2+(8-y)) = 52 (solve for y) y = 5

Pumping alone at their respective constant rates, one inlet pipe fills an empty tank to 1/2 of capacity in 3 hours and a second inlet pipe fills the same empty tank to 2/3 of capacity in 6 hours. How many hours will it take both pipes, pumping simultaneously at their respective constant rates, to fill the empty tank to capacity?

Combined rate problem 1/r + 1/t = 1/combined (combined rates) Hours - reciprocal of rate Pipe 1: 3r = 1/2 -- r = 1/6 Pipe 2: 6r = 2/3 -- r = 1/9 Combined rate = 1/6 + 1/9 = 5/18 time = 1/(5/18) = 18/5 = 3.6 hrs

20 copies/min How many hours to get 4800 copies?

Convert to hours! 20 copies/min * (60 min / 1 hr) = 1200 copies/hr 1200x = 4800 x = 4

If (1 - 1.25)N = 1, then N =

Look at problem carefully! -1/4N = 1 N = -4

Al and Ben are drivers for SD Trucking Company. One snowy day, Ben left SD at 8:00 a.m. heading east and Al left SD at 11:00 a.m. heading west. At a particular time later that day, the dispatcher retrieved data from SD's vehicle tracking system. The data showed that, up to that time, Al had averaged 40 miles per hour and Ben had averaged 20 miles per hour. It also showed that Al and Ben had driven a combined total of 240 miles. At what time did the dispatcher retrieve data from the vehicle tracking system?

D = Rate x Time Put t in the same timeframe A Distance = 40 (t+3) B Distance = 20t Total Distance = 40 (t+3) + 20t = 240 40t + 120 + 20t = 240 60t = 360 t = 6 8 am + 6 hours = 2 pm

Working simultaneously and independently at an identical constant rate, 4 machines of a certain type can produce a total of x units of product P in 6 days. How many of these machines, working simultaneously and independently at this constant rate, can produce a total of 3x units of product P in 4 days?

Define a machine day as 1 machine working for 1 day. Then, 4 machines each working 6 days is equivalent to (4)(6) = 24 machine days. Thus, x units of product P were produced in 24 machine days, and 3x units of product P will require (3)(24) = 72 machine days, which is equivalent to 72/4 = 18 machines working independently and simultaneously for 4 days.

The cost C, in dollars, to remove p percent of a certain pollutant from a pond is estimated by using the formula C = 100,000p/(100-p). According to this estimate, how much more would it cost to remove 90 percent of the pollutant from the pond than it would cost to remove 80 percent of the pollutant?

Do out the 2 calculations for 90% and 80% 90% = 9,000,000/10 = 900,000 80% = 8,000,000/20 = 400,000 Difference = 500,000

Quant Review DS 261 Distance btw A and C? 1) distance btw A and B = 75 2) distance btw B and C = 135

Don't assume straight line! Could be on a triangle / whatever shape 1) NS (no info on C) 2) NS no info A Both: doesn't have enough info on whether straight line or what type of shape E

Tanks A and B are each in the shape of a right circular cylinder. The interior of Tank A has a height of 10 meters and a circumference of 8 meters, and the interior of Tank B has a height of 8 meters and a circumference of 10 meters. The capacity of Tank A is what percent of the capacity of Tank B ?

Find Volume A and Volume B (calculate things out!) Volume of right cylinder = pi*r^2*h Volume A C = 8 which means r = 4/pi Volume = pi*(4/pi)^2 * 10 = 160/pi Volume B C = 10 which means r = 5/pi Volume = pi*(5/pi)^2 * 10 = 200/pi 160/pi/250/pi = 160/200 = 16/20 = 4/5 = 80%

If mn ≠ 0 and 25 percent of n equals 37.5 percent of m, what is the value of 12n/m?

Find n/m and then multiply by 12 .25n=0.375m n/m = 0.375/0.25 (3/8/1/4) 3/2 *12 = 18

Which of the following fractions is closest to 1/2 ? 4/7 5/9 6/11 7/13 9/16

Find the distance between numbers and 1/2 and smallest difference is the closest 4/7 - 1/2 = 1/14 5/9 - 1/2 = 1/18 6/11 - 1/2 = 1/22 7/13 - 1/2 = 1/26 9/16 - 1/2 = 1/16 Alternatively, double numbers and find which one is closest to 1 8/7 = 1 + 1/7 5/9 = 10/9 = 1 + 1/9 6/11 = 12/11 = 1 + 1/11 7/13 = 14 /13 = 1 + 1/13 9/16 = 18/16 = 9/8 = 1 + 1/18

Team needs to make an average of 200 items per day in September. For first half of month, team produced 150 items/day. How many items per day must team average during second half of the month to attain the work plan?

First half shortfall = 200-150 = 50 items/day Second half must make up for this shortfall on top of the required 200 items / day Thus items produced must be 50 + 200 = 250 items/day

Testing values in DS What values? FROZEN

Fractions Roots One Zero Extremes Negative

If x and y are positive numbers such that x + y = 1, which of the following could be the value of 100x + 200y ? 80 140 199

Know the constraint of x and y being positive numbers. Test each value! Since x + y = 1, then y = 1 − x and 100x + 200y can be expressed as 100x + 200(1 − x) = 200 − 100x. Test each value. If 200 − 100x = 80, then x = (200−80)/100 = 1.2 and y = 1 − 1.2 = −0.2. Since y must be positive, 80 cannot be a value of 100x + 200y. If 200 − 100x = 140, then x = (200−140)/100 = 0.6 and y = 1 − 0.6 = 0.4, so 140 can be a value of 100x + 200y. If 200 − 100x = 199, then x = (200−199)/100 = 0.01 and y = 1 − 0.01 = 0.99, so 199 can be a value of 100x + 200y.

Last year a state senate consisting of only Republican and Democrat members had 20 more Republican members than Democrat members. This year the senate has the same number of members as last year, but it has 2 fewer Republican members than last year. If this year the number of Republican members is 2/3 the number of senate members, how many members does the senate have this year?

Last year: R = D+20 Total = 2D + 20 This year: R = D+18 D = D+2 R = 2/3 of total D+18 = 2/3 (2D+20) [multiply by 3) 3(D+18) = 2(2D+20) 3D + 54 = 4D + 40 14 = D R = 14+18 = 32 Total = 2D + 20 = 2(14) + 20 = 28+20 = 48

Quant review PS #29 For positive integers a and b, the remainder when a is divided by b is equal to the remainder when b is divided by a. Which of the following could be a value of ab? I) 24 II) 30 III) 36

Need to know what this implies for value of a and b 2 conditions: a=b or a does not equal b Condition 1: a = b means a/b = a/a b/a = a/a Works Condition 2: a does not equal b a<b --> a/b = remainder is a (a=7, b=10 means remainder is 7 for example) b/a = remainder is not a (10/7 = remainder is 3 now) Does not work given the conditions Now consider the statements knowing that a=b I) 24 = 3*2*2*2 --> does not work because only one 3 and can't make a*a = 24 II) 30 = 5*3*2 - no integer that makes a*a = 30 III) 36 = 6*6 works

The sum of all the integers k such that −26 < k < 24 is

Notice that this is asking -25 to 23 and pairing 23 and -23 gets 0 sum Sum = −25 + (−24) + (23)(0) = −49

Which is the least out of the fractions? 0.03/0.00071 0.03 /0.0071 0.03/071 0.03/0.71 0.03/7.1

Numerator is the same same so the least will be when the denominator is largest - fraction rule! As denominator increases, fraction gets smaller!

If n is a positive integer, is n odd? (1) 3n is odd. (2) n + 3 is even.

Odd and even rules Make sure to look at what makes each statement is true 1) Odd * (n) = Odd n = Odd (only possibility) 2) n + Odd = Even n = Odd

6(87.30 + 0.65) - 5(87.30) =

Recognize the same multiplier and can distribute 6(87.30) + 6(0.65) - 5(87.30) 6-5 (87.3) +6(0.65) 87.3 + 3.9 91.2

The price of gasoline at a service station increased from $1.65 per gallon last week to $1.82 per gallon this week. Sally paid $26.40 for gasoline last week at the station. How much more will Sally pay this week at the station for the same amount of gasoline?

The amount that Sally purchased last week is equal to the total price that Sally paid last week divided by the price per gallon last week. Since gallons is the amount that Sally purchased this week at a price of $1.82 per gallon, the amount that Sally paid this week , which is more than Sally paid last week.$0.17*(26.4/1.65) = 2.72

Bouquets are to be made using white tulips and red tulips, and the ratio of the number of white tulips to the number of red tulips is to be the same in each bouquet. If there are 15 white tulips and 85 red tulips available for the bouquets, what is the greatest number of bouquets that can be made using all the tulips available?

Topic: divisibility / factors White tulips must be a factor of 15 = 1, 3, 5, 15 Red tulips must be a factor of 85: 1, 5, 17, 85 Number of bouquets must equal 1 or 5 Max bouquets = 5

x+y = 2 X^2 + y^2 = 2 What is xy?

Trick - doing something with what's given will land xy square x+y x^2 + 2xy + y^2 = 4 (can substitute values) 2xy + 2 = 4 2xy = 2 xy = 1

If n = (33)^43 + (43)^33, what is the units digit of n ?

Units Digit property - can just look at units Notice patterns of power of 3 Remember to add the unit digits! If the units digit of an integer n is 3, then the units digits of n1, n2, n3, n4, n5, n6, n7, and n8 are, respectively, 3, 9, 7, 1, 3, 9, 7, and 1. Thus, the units digit of the powers of n form the sequence in which the digits 3, 9, 7, and 1 repeat indefinitely in that order. Since 43 = (10)(4) + 3, the 43rd number in the sequence is 7, and therefore the units digit of (33)43 is 7. Since 33 = (8)(4) + 1, the 33rd number of this sequence is 3, and therefore, the units digit of (43)33 is 3. Thus, the units digit of (33)43 + (43)33 is the units digit of 7 + 3, which is 0.

If a(a + 2) = 24 and b(b + 2) = 24, where a ≠ b, then a + b =

factor out and solve for solutions (no easy way to get a+b) a^2+2a-24=0 (a+6)(a-4) = 0 a = -6 or 4 b^2+2b-24=0 (b+6)(b-4) = 0 b = -6 or 4 -6+4 = -2

If each of the 12 teams participating in a certain tournament plays exactly one game with each of the other teams, how many games will be played?

no repeats so combination 12!/2!10! = 12*11*10!/10!2 = 66

Clarissa will create her summer reading list by randomly choosing 4 books from the 10 books approved for summer reading. She will list the books in the order in which they are chosen. How many different lists are possible?

Order does matter so permutation 10*9*8*7 = 5040

If of the total number of marbles in the three bags listed in the table above are blue, how many marbles are there in bag Q ? Number of Marbles in Each of Three Bags Percent of Marbles in Each Bag That Are Blue (to the nearest tenth) Bag P 37 10.8% Bag Q × 66.7% Bag R 32 50.0%

P=37*0.108 = 4 Q = 2/3 x R = 16 Total Blue = 20 + 2x/3 1/3 (37+32+x) = 20 + 2x/3 x/3 + 23 = 20 + 2x/3 x = 9

A worker carries jugs of liquid soap from a production line to a packing area, carrying 4 jugs per trip. If the jugs are packed into cartons that hold 7 jugs each, how many jugs are needed to fill the last partially filled carton after the worker has made 17 trips?

READ THE QUESTION! Note it's asking about jugs 17*4 = 68 Total cartons = 7*10 = 70 Jugs needed = 70-68 = 2

Three printing presses, R, S, and T, working together at their respective constant rates, can do a certain printing job in 4 hours. S and T, working together at their respective constant rates, can do the same job in 5 hours. How many hours would it take R, working alone at its constant rate, to do the same job?

Rate*Time = Work 1/a + 1/b +1/c = 1/time together (Rate) Let r be the portion of the job that printing press R, working alone, completes in 1 hour; and let s and t be the corresponding portions, respectively, for printing press S and printing press T. From the given information, it follows that r+s+t=1/4 and s+t=1/5. Subtracting these two equations gives r=1/4−1/5=1/20. It follows that printing press R, working alone, will complete 1/20 of the job in 1 hour, and therefore printing press R, working alone, will complete the job in 20 hours.

If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following? 3 16 17 18 10

Since it is given that y is a multiple of 5, let y = 5q, where q is a positive integer. It is given that 3x + 4y = 200, so 3x + 4(5q) = 200 or 3x + 20q = 200. It follows that 3x = 200 - 20q = 20(10 - q). Since 3 is prime and is not a factor of 20, then 3 must be a factor of 10 - q, which means the only possible values of q are 1, 4, and 7. This is summarized in the following table. In each case, the value of x is a multiple of 10. 1. Alternatively, since 3x = 20(10 - q), the factors of the product of 3 and x must correspond to the factors of the product of 20 and (10 - q). Since the factors of 20 include 10, then x must have a factor of 10, which means that x is a multiple of 10.

Exchange Rates in a Particular Year $1 = 5.3 francs $1 = 1.6 marks An American dealer bought a table in Germany for 480 marks and sold the same table in France for 2,385 francs. What was the dealer's gross profit on the two transactions in dollars?

Use division 480 marks = $300 2385 francs = $450 Difference = $150

A certain state's milk production was 980 million pounds in 2007 and 2.7 billion pounds in 2014. Approximately how many more million gallons of milk did the state produce in 2014 than in 2007 ? (1 billion = 109 and 1 gallon = 8.6 pounds.)

Using 2.7 billion = 2,700 million, the amount of increase in millions of gallons can be calculated as follows. 1. The amount of increase is: (2,700 - 980) million pounds = 1,720 million pounds 2. Convert to gallons 1720 M *8.6 = 200M

230 Quant Review DS How many hours would it take pump A and pump B working together each at its own constant rate to empty a tank that was initially full? 1) Working alone at its constant rate Pump A would empty the full tank in 4 hours 20 minutes 2) Working alone Pump B would empty the full tank at its constant rate of 72 liters per min

W = R*T 1) time for A but no info on B's constant rate - NS 2) Gives B's rate but no information on A capacity or rate Both: time for A and rate for B - still not enough info to find time working together To get the time, you need the rate for both and because we don't have A's capacity, we can't figure out the rate

After driving to a riverfront parking lot, Bob plans to run south along the river, turn around, and return to the parking lot, running north along the same path. After running 3.25 miles south, he decides to run for only 50 minutes more. If Bob runs at a constant rate of 8 minutes per mile, how many miles farther south can he run and still be able to return to the parking lot in 50 minutes?

total amount of time running - 50 3.25 miles at 8 miles/min = 26 min already done 50-26 = 24 min left to run forward and back 12 min /8 = 1.5 miles

x>y y>z What is largest difference?

x>y>z x-z Recognize the values and put in actual values if needed to see it!