Word Problems

Question 1

Age PB
Right now, Steve’s age is half of Tom’s age. In eight years, twice Tom’s age will be five more than three times Steve’s age. How old is Tom right now?

Answer 1

Pick the variable to represent the age right now
T = 2S
2(T + 8) = 5 + 3(S + 8)
2(2S + 8) = 3(S + 8) + 5
4S + 16 = 3S + 24 + 5
S = 13
T = 26

Question 2

Motion PB formula

Answer 2

D = RT
D : distance
R : rate
T : time

Question 3

A lichen advances 4cm each year across a rock slab. If this rate remains constant over time, how many years will it take to cross 30m? (1m = 100cm)

Answer 3

R = 4cm/yr
D = 30m = 3000cm
T = D/R = 3000cm/4cm/yr = 750yrs

Question 4

A car moving at 72km/hr moves how many meters in one second? (1km = 1000m)

Answer 4

T = 1hr = 3600s
R = 72 000m / 3600s = 20 m/s

Question 5

Bob drove 120 miles at 60 mph, then another 120 at 40 mph, What was his average speed for the total trip?

Answer 5

average velocity = total distance / total time
D = 120mi, T = D/R = 120 mi / 60 mph = 2hr
D = 120mi, T = D/R = 120 mi / 40 mph = 3hr
V = D/T = 240mi / 5hr = 480mi / 10hr = 48mph

Question 6

Cassandra drove from A to B at a constant 60 mph speed. She then returned, on the same route, from B to A, at a constant speed of 20mph. What was her average speed?

Answer 6

D = distance from A to B
T1 = D/R = D/60
T2 = D/R = D/20
Tt = T1 + T2 = D/60 + D/20 = 4D/60 = D/15
Vavg = Dt/Tt = 2D/(D/15) = 2D/1 x 15/D = 30mph

Question 7

An airplane has a 3600-mile trip. It covers the first 1800 miles of a trip at 400 mph. Which of the following is the closest to the constant speed the plane would have to follow in the last 1800 miles so that the average speed of the whole trip is 450 mph?
A) 450 mph
B) 455 mph
C) 500 mph
D) 514 mph
E) 600 mph

Answer 7

1st leg : D = 1800 R = 400
T = D/R = 1800/400 = 4.5

whole trip : D = 3600, R = 450
T = D/R = 3600/450 = 8hr

2nd leg : 8 - 4.5 - 3.5hr

T = D/T = 1800/3.5 = 3600/7

Estimate
3600/7 > 3500/7 = 500mph
3600/7 < 3600/6 = 600 mph

OR

Calculate
3600/7 = 3500+100 / 7 = 3500/7 + 100/7
= 500 + 100/7
That’s about 514

Answer D

Question 8

Martha and Paul started traveling from A to B at the same time. Martha traveled at a constant speed of 60 mph, and Paul at a constant speed of 40 mph. When Martha arrived at B, Paul was still 50 miles away. What is the distance between A and B?

Answer 8

D = 60T
D - 50 = 40T
60T -  50 = 40T
20T = 50
T = 2.5hr

D = 60 x 2.5 = 150mi

Question 9

Frank and Georgia started traveling from A to B at the same time. Georgia’s constant speed was 1.5 times Frank’s constant speed. When Georgia arrived at B, she turned around immediately and returned by the same route. She crossed paths with Frank, who was coming toward B, when they were 60 miles away from B. How far away are A and B?

Answer 9

D - 60 = RT
D + 60 = 1.5RT
D + 60 = 1.5(D - 60)
D + 60 = 1.5D - 90
60 = 0.5D - 90
150 = 0.5D
D = 300

Question 10

Multiple Travelers Questions Strategy

Answer 10

each traveler must have its own D = RT formula
try to express one variable in terms of another by substitution

two travelers moving in OPPOSITE directions, ADD the speeds

• two travelers approaching each other: the sum of the speeds = speed at which the gap is shrinking
• two travelers moving away from each other: the sum of the speeds = speed at which the gap is expanding

two travelers moving in the SAME direction, SUBTRACT the speeds

• faster traveler in front: difference in speeds = speed at which the gap is expanding
• slower traveler in front: difference in speeds = speed at which the gap is shrinking

sometimes it saves time to set up a D = RT for the gap itself

Question 11

A car and a truck are moving in the same direction on the same highway. The truck is moving at 50 mph, and the car is traveling at a constant speed. At 3:00 pm, the car is 30 miles behind the truck and at 4:30 pm, the car overtakes and passes the truck. What is the speed of the car?

Answer 11

GAP : R = D/T = 30/1.5 = 20mph

20mph = difference in speeds, so the car must be moving at 50 + 20 = 70mph

Question 12

Car X and Y are traveling from A to B on the same route at constant speeds. Car X is initially behind ar Y, but Car X’s speed is 1.25 times Car Y’s speed. Car X passes Car Y at 1:30 pm. At 3:15 pm, Car X reaches B, and at that moment, Car Y is still 35 miles away from B. What is the speed of Car X?

Answer 12

Time interval: 1:30pm to 3:15pm =1 + 3/4h = 7/4hr
GAP: R = D/T = 35mi/(7/4)hr = 140/7 = 20mph
Let x = speed of Car X and y = speed of Car Y
x = 1.25y
x - y = 20
1.25y - y = 20
0.25y = 20
y = 80mph
x = 80+20 = 100mph

Question 13

Cars P & Q are approaching each other on the same highway. Car P is moving at 49 mph and Car Q is moving at 61 mph. At 2:00 pm, they are approaching each other and 121 mi apart. Eventually, they pass each other. At what clock time are they moving away from each other and 44 miles apart?

Answer 13

Gap changing at R = 49 + 61 = 110 mph
approaching gap = 121 mi
receding gap = 44 mi
total gap = 165 mi
T = D/R = 165/110 = 15/10 = 1.5hr
3:30 pm

Question 14

Work Equation

Answer 14

A = RT

A = amount of work done (products manufactured, houses painted, pizzas made)
R = work rate (nb per mn, per day, per hour)
T = time

different machines or workers working at different work rates: combined work rate of people/machines working together = sum of the individual work rates
we have to construct rates and add/subtract them

Question 15

A machine, working at a constant rate, manufactures 36 staplers in 28mn. How many staplers does it make in 1hr 45mn?

Answer 15

1 hr 45 mn = 105 min
staplers / time = 36/28 = S / 105
9/7 = S/105
9/1 = S/15
S = 9*15 = 135

Question 16

To detail a car means to clean it thoroughly, inside and out. When Amelia and Brad detail a car together, 1 car takes 3 hours. When Amelia details a car alone, 1 car takes 4 hours. How long does it take Brad, working alone, to detail one car?

Answer 16

rates = cars/hours
Rab = 1/3
Ra = 1/4
Ra + Rb = Rab
Rb = Rab - Ra
Rb = 1/3 - 1/4
Rb = 1/12
12 hours

Question 17

Pump X takes 28 hours to fill a pool. Pump Y takes 21 hours to fill the same pool. How long does it take them to fill the same pool if they are working simultaneously?

Answer 17

rate = pool / hour
Rx = 1/28
Ry = 1/21
R = Rx + Ry
R = 1/28 + 1/21
R = 4/84 + 3/84
R = 7/84 = 1/12
12 hours

Question 18

When isotope QXW radioactively decays, it loses exactly half of its mass in each three-day period. Suppose scientists start with a 96-gram sample of pure isotope QXW on a certain day. What will be the remaining mass in 12 days?

Answer 18

start : 96g
\+3days : 48
\+3days : 24
\+3days : 12
\+3days : 6

Question 19

Under optimal conditions, the Vericoccus Bacteria multiplies the size of its population by 5/2 every 4 hours. If there are 24 billion at 9:00 am, and optimal conditions are maintained, how many are there at 5:00pm of the same day?

Answer 19

9 am: 24 billion
1 pm: 24 x 5/2 = 60 b
5 pm: 60 x 5/2 = 150 b
150 billion

Question 20

Concentration formula

Answer 20

concentration = ( amount of solute / total amount of solution ) x 100
notice that solution = solvant + solute

Question 21

How much HCI and how much water must we use to create 5 liters of a 30% HCI solution?

Answer 21

amount of HCl = 0.3 x 5 = 1.5L

amount of water = 5 - 1.5 = 3.5L

Question 22

Suppose we start with 5 liters of a 30% HCl solution. How much water must we add to create a 20% solution?

Answer 22

amount of HCl = 0.3 * 5 = 1.5L

new solution:
amount of HCl = 0.2 * X = 1.5L
1/5 * X = 1.5
X = 7.5L total
added water = 7.5 - 5 = 2.5

Question 23

Suppose we start with 8 liters of a 60% H3PO4 solution. We add 4 liters of C% H3PO4 solution, and the result is 12 liters of a 50% H3PO4 solution. What is C?

Answer 23

solute in 1st solution = 0.6 * 8 = 4.8L
solute in result = 0.5 * 12 = 6L
solute added = 6 - 4.8 = 1.2L

C = 1.2/4 = 0.3 = 30%

Question 24

Suppose we start with unlimited supplies of a 20% H2SO4 solution and of a 50% H2SO4 solution. We combine X liters of the first with Y liters of the second to produce 7 liters of a 40% H2SO4 solution. What does X equal?

Answer 24

X + Y = 7
solute = 0.4 * 7 = 2.8L
solute 1 = 0.2 X
solute 2 = 0.5 Y
0.2X + 0.5Y = 2.8

Y = 7 - X
0.2X + 0.5(7-X) = 2.8
2X + 5(7-X) = 28
2X + 35 - 5X = 28
35 - 3X = 28
3X = 7
X = 7/3L

Question 25

At a certain school of 200 students, the students can study French, Spanish, both or neither. Just as many study neither as study both. One quarter of those who study Spanish also study French. The total number who study French is 10 fewer than those who study Spanish only. How many students study French only?

Answer 25

Venn Diagramm
Left circle = French (A+B)
Right circle = Spanish (C+B)
Both = B
Neither = D

B = D
1/4 (B+C) = B
B+C = 4B
C = 3B

A + B = C - 10
A + B = 3B - 10
A = 2B - 10

200 = A + B + C + D
200 = (2B - 10) + B + 3B + B
210 = 7B
B = 30
A = 2*30 - 10
A = 50

Question 26

There are a total of 400 students at a school, which offers a chorus, baseball, and Italian. This year, 120 students are in the chorus, 40 students in both chorus & Italian, 45 students in both chorus & baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how many students are in none of the three activities?

Answer 26

3-way Venn Diagram
T = 400
C + E + F + G = 120
E + G = 40
G + F = 45
G = 15

Italian or Baseball = A + B + D + E + F + G = 220
E = 25
F = 30
C = 50

A + B + D + E + F + G + C = 270
H = 400 - 270 = 130

Question 27

Basic patterns
sequence of all positive integers
sequence of all positive odd nb
sequence of all positive multiples of 7
sequence of all perfect squares
sequence of all powers of 3

Answer 27

a(n) = n
a(n) = 2n - 1
a(n) = 7n
a(n) = n²
a(n) = 3^n

Question 28

Which of the following could be true of at least some of the terms of the sequence defined by
b(n) = (2n - 1)(2n + 3)

I. divisible by 2
II. divisible by 3
III. divisible by 5

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II and II

Answer 28

b(1) = 15 = 5
b(2) = 37 = 21
factor of odds so odd result
(D) II and III only

Question 29

Find the 41st term of this sequence

14, 23, 32, 41, 50, 59, …

Answer 29

a(1) = 14
d = 9

a(n) = 14 + 9(n - 1)
a(41) = 14 + 9(40) = 374

Question 30

Let S be the set of all positive integers that, when divided by 8, have a remainder of 5. What is the 76th nb in this set?

Answer 30

a(1) = 5
d = 8

a(n) = 5 + 8(n - 1)
a(76) = 5 + 8(75) = 5 + 4*150 = 605

Question 31

Let T be a sequence of the form
a(n) = a(1) + d(n - 1)
If a(3) = 17 and a(19) = 65, find a(10)

Answer 31

a(19) = a(1) + d(n-1) = a1 + 18d = 65
a(3) = a(1) + 2d = 17

subtract the two equations :
16d = 48
d = 3
a(1) = 11

a(10) = 11 + 3*9 = 38

Question 32

nth term of any arithmetic sequence formula
definition arithmetic sequence
definition recursive sequence

Answer 32

a(n) = a(1) + d*(n-1)

an arithmetic sequence is one in which the terms have a common difference
any evenly spaced list is an arithmetic sequence
(consecutive multiples of a nb, consecutive odds or evens, same remainder when divided by the same divisor, …)

in recursive sequences, each term a(n) is defined in terms of one or two previous terms a(n-1), a(n-2)
no way to jump to the value of term: term by term

Question 33

How many multiples of 8 are there from 200 to 640 inclusive?

Answer 33

200 = 8*25
640 = 8*80
200 is the 25th multiple of 8
640 is the 80th multiple of 8
200 and 640 are both included
nb = 80 - 25 + 1 = 56
(inclusive counting: +1)

Question 34

What is the sum of all the multiples of 20 from 160 to 840?

Answer 34

To add sequences of evenly-spaced nb, pair the nb by first and last to produce pairs of a constant sum; then multiply by the nb of pairs :
sum of list = N(a1 +aN) / 2

160 = 8*20
640 = 42*20
42 - 8 + 1 = 35
nb of pairs = 17.5
sum of list = 17.5(160+840) = 17,500

Question 35

If n is an integer greater than 10, then the expression (n² - 2n)(n + 1)(n - 1) MUST be divisible by which of the following?
I. 4
II. 6
III. 18

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II and III

Answer 35

(n² - 2n)(n + 1)(n - 1)
n(n - 2)(n + 1)(n - 1)
(n - 2)(n - 1)n(n + 1)

4 consecutive integers = 2 evens and 2 odds
2 evens = divisible by 4

4 consecutive integers = at least one integer is divisible by 3
1 even + 1 divisible by 3 = divisible by 6

if we take examples of 4 consecutive integers, we could have two factors of 3 (3, 4, 5, 6) or only one factor of 3 (4, 5, 6, 7)
not ALWAYS divisible by 18

(C) I and II only

Question 36

N = 135 is the lowest of a set of 11 consecutive multiples of 5. What is the difference between the highest and lowest nb?

Answer 36

50

Question 37

In a certain state, schools pay 2% tax on food and 8% on stationery. A school placed a combined order of $500 on food and stationery, and paid $19 on tax on the order. How much of that money was spent on food?
(A) $200
(B) $250
(C) $300
(D) $350
(E) $400

Answer 37

(C) $300 on food, $200 stationery
food tax = $6
stationery tax = $16
total tax = $22
to pay less in taxes: spend more on food, less on stationery

eliminate A, B and C

solve for D and E
answer (D) $350

Question 38

A chemical supply company has 60 liters of a 40% HNO3 solution. How many liters of pure undiluted HNO3 must the chemists add so that the resultant solution is a 50% solution?
A 12
B 15
C 20
D 24
E 30

Answer 38

(C) 20 L added
total concentrate = 24 + 20 = 44L
total solution = 60 + 20 = 80L
44L is more than half of 80L, too high

Eliminate (C) (D) and (E)

(A) 12L added
total concentrate = 24 + 12 = 36L
total solution = 60 + 12 = 72L
36 is half of 72 OK

Question 39

Jennifer can buy watches at a price of B dollars per watch, which she marks up by a certain percentage before selling. If she makes a total profit of T by selling N watches, then in terms of B and T and N, what is the percent of the markup from her buy price to her sell price?

(A) 100T/(NB)
(B) TB/(100N)
(C) 100TN/B
(D) ((T/N) - B)/(100B)
(E) 100(T - NB)/N

Answer 39

Algebraic approach:
B is the buy price
let S be the sell price
profit per watch = S - B
for N watches, profit = N(S - B) = T
(S - B) = T / N
the % increase = (S - B) / B   x 100 = 100(T/N) / B = 100T / NB
(A)

Plug-in approach:
price per watch = B = $10
mark-up percent 30%
sell price = $13
profit per watch = $3
If she sells N = 20 watches, T = $60

if we plug values of B=10, N=20, T=60 into the answer choices, the correct answer will yield a value of 30
(A)

Question 40

At the store, Sam bought a shirt and a toaster. There was an 8% sales tax on each item, and with tax, Sam paid a total of Q. If the price of the toaster before tax was T, what, in terms of Q and T, is the price of the shirt?
(A) 0.92(Q - T)
(B) 0.92Q - T
(C) 0.92(Q - 1.08T)
(D) (Q - T) / 1.08
(E) (Q/1.08) - T

Answer 40

Let S be the shirt price, and T the toaster price, both before taxes. The bill before taxes is S + T. With 8% tax added, this is Q = 1.08(S + T)
To solve for S, we have to undo an 8% increase.
!!! an 8% decrease does NOT undo an 8% increase.
The answer choices with 0.92 (8% decrease) are all incorrect
eliminate A, B, and C
solve the equation for S
1.08(S+T) = Q
S + T = Q/1.08
S = (Q/1.08) - T
(E)

Question 41

For the first leg of a trip, Fred traveled A miles at speed p.
During the second leg, he traveled at a slower speed.
There were only two legs in the trip.
The entire trip took T hours, and the average speed for the entire trip was V. In terms of A, p, T, and V, what was the average speed of the second leg of the trip?*
(A) (VT - A) / T
(B) (V - p) / pT
(C) (V - A/T) / (1 - A/pT)
(D) (V - A/T) / (V - p)

Answer 41

First leg
Distance = A
Speed = p
Time = A/p

Whole trip
Speed = V
Time = T
Distance = VT

Second leg
Distance = VT - A
Time = T - A/p
Speed = (VT - A) / (T - A/p)

Divide by T:
(VT - A) / (T - A/p) = (V - A/T) / (1 - A-pT)
Answer (C)

Question 42

Before January, the price of a dress was D and the price of a matching pair of shoes was H.
In January, the price of the dress increased by 40% and the price of the shoes increased by 50% and didn’t change again in the following months. In March Roberta bought both items with a 30% discount. If D = 5H, which of the following represents the amount that Roberta paid?

(A) D +40
(B) D + H - 1
(C) D + 2H
(D) 5.95H
(E) 1.21D

Answer 42

D