Word Problems Flashcards
Put into an equation…
x less than 3
3 - x
Put into an equation…
x less 3
x - 3
Put into an equation…
Up to 4
≤ 4
Put into an equation…
4 or less
≤ 4
Put into an equation…
At least 4
≥ 4
Put into an equation…
4 or more
≥ 4
Put into an equation…
A times B
A x B
Put into an equation…
The quotient of A and B
A / B
Put into an equation…
P increased by 7
P + 7
Put into an equation…
P decreased by 7
P - 7
Put into an equation…
One-third the difference of A and B
(1/3)(A - B)
Put into an equation…
3/4 of Q
3/4 x Q
Put into an equation…
Is the same as
=
How would you write the equation for….
A number is six times another number. Their sum in 28. What is the smaller number?
x + 6x = 28
How would you write the equation for….
Twice the length of a cardboard box is three times its width. If the length and width together span 15 inches, what is the width?
Length = 15 - x
2(15 -x) = 3x
How would you write the equation for….
A beauty shop sold 200 bottles of shampoo for $1250. One type of shampoo sells for $5; the other is twice as much. How many of the higher priced bottles were sold?
200 - x = Number of $5 bottles
5(200 - x) + 10x = 1250
What are consecutive integers?
-2, -1, 0, 1, etc.
What are consecutive odd integers?
-3, -1, 0, 1, etc.
What are consecutive even integers?
-2, 0, 2, 4, etc.
What are three consecutive integers whose product is 0?
3
0, 2, 4
- 2, 0, 2
- 4, -2, 0
How would you write the equation for….
A car dealership greets 500 visitors each month. 2/3 of the visitors drive a car. 2/5 of the test drivers do not buy. No one buys a car without test driving. How many cars does the dealership sell each month?
500 x 2/3 x 3/5
A salary was increased last year from $20,000 to $25,000. If the salary is increased this year by the same percentage, what will be the new salary?
25,000 - 20,000 = 5,000
5,000 / 20,000 = 0.25
25,000 x 1.25 = 31.250
How would you write “if a number was increased by 5%”?
x 1.05
How would you write “if a number is decreased by 10%”?
x 0.90
How would you write “if a number is decreased by 40% then raised by 20%”?
(# x 0.60) x 1.20
A product is reduced by 20% and now costs $5. What was the original price?
5 = p - 0.2p
5 = 0.8p
6.25 = p
A store wants to make a 25% profit on cards that cost the store $4. What price should be set?
c + 0.25c = 1.25c
I miss 3 monthly payments. What percent do I need to increase my next 9 payments in order to catch up by the end of the year?
9m = 12p
m = 12p/9
m = 4p/3
p = 1.33
A thunderstorm is 10% the current flying altitude. The pilot then increases the flying altitude by 10%. Now the plane is 6,000 feet above the thunderstorm. How high above the thunderstorm was the plane originally?
Thunderstorm = 0.9a
New Altitude = 1.1a
Difference = 0.2a
0.2a = 6000
a = 6000/0.2
a = 30,000
I sold my TV for $100 taking a $25 loss. What percent was my loss?
Original cost of TV = 100 + 25 = 125
25/125 = 20%
I bought a car for $1000. I want to make a 40% profit but the website requires me to list the car for 30% below asking price. At what price should I list the car?
(Desired Price) 1.4 x 1000 = 1400
(Asking Price) 0.7p = 1400
p = 2000
A dealer sells blue and yellow chairs at the same price. Blue chairs sell so well that the owner increases their cost by 10%. Yellow chairs do not sell as well so the owner decreases their cost by 20%. The difference in the new prices is $12. What was the original price of the chairs?
Blue = 1.1p Yellow = 0.8p
12 = 1.1p - 0.8p 12 = 0.3p p = 12/0.3 p = 120/3 p = 40
A house loses 30% of its value. If the house is to retain its original value, to what percent would its value now need to increase?
New value = 0.7h
0.3h/0.7h x 100
3h/7h x 100
300h/7h
42.86
What are directly proportion ratios?
Ones where an increase (or decrease) in one quantity results in an increase (or decrease) in another quantity
What are indirectly proportional ratios?
Ones where an increase (or decrease) in one quantity causes an decrease (or increase) in another quantity
Two clubs (A & B) had 20 & 10 members respectively. Last month, Club B acquired twice as many members as Club A. The ratio of members of Club A to Club B is now 3:5. How many new members joined Club B?
(20 + x) : (10 + 2x) = 3:5
(20 + x)/(10 + 2x) = 3/5
5(20 + x) = 3(10 + 2x)
100 + 5x = 30 + 6x
70 = x
The ratio of vegetarians to carnivores is 13:4. What fraction of the population eats meat?
(Total Population) 13 + 4 = 17
4/17
The sum of 20 numbers is what percent of their average?
20 x 100
2,000
The average of 4 numbers is 20. If you increase one number by 8 and decrease another by 4, what will be the new average?
20 x 4 = 80
80 + 8 - 4 = 84
84/4 = 21
A charity received 10 donations worth either $5 or $20. If the average donation was $14, how many $5 donations were there?
5d + 20(10-d)/10 = 14
5d + 20(10-d) = 140
5d + 200 - 20d = 140
60 = 15d
4 = d
Calculate the dollar value of a mixture consisting of 100 barrels of oil that costs $50 per barrel and 250 barrels of oil that costs $120 per barrel.
50(100) + 120(250)
5,000 + 30,000
35,000
A 200 pound cereal mixture costs $3.25/lb. Corn flakes are $4.00/lb. Wheat flakes are $1.50/lb. How many pounds of corn flakes are in the mix?
4x + 1.5(200-x)/200 = 3.25
4x + 1.5(200-x) = 650
4x + 300 + 1.5x = 650
2.5x = 350
x = 140
5 gallons of fuel consist of 80% alcohol and 20% gasoline. How many gallons of alcohol must be added to change the composition to 90% alcohol and 10% gas?
(4 + x) / (5 + x) = 0.9
4 + x = 0.9(5 + x)
4 + x = 4.5 + 0.9x
0.1x = 0.5
x = 5
What is the best strategy for solving age related problems?
Make a chart
What does “X years from now” mean?
X is added to the variable
What does “X years ago” mean?
X is subtracted from the variable
Four years ago, Pam was twice Sam’s age at the time. How old is Pam now?
P - 4 = 2(P-4)
P - 4 = 2P - 8
4 = P
Veronica is 7x as old as her daughter. In four years, Veronica will be 4x as old as her daughter. How old is Veronica today?
Daugher Veronica
Now d 7d
+ 4 yrs d + 4 7d + 4
7d + 4 = 4(d+4)
7d + 4 = 4d + 16
3d = 12
d = 4
Colleen is twice as old as Liz today. Five years from now, twice Colleen’s age will be equal to 3x Liz’s age. How old was Colleen back then?
Colleen Liz
Now 2L L
+5 yrs 2L + 5 L + 5
2(2L+5) = 3(L + 5)
4L + 10 = 3L + 15
L = 5
Peter is six years older than Rebecca. 18 years ago, he was twice her age back then. How old is Rebecca today?
Rebecca Peter
Now R R+6
-18yrs R - 18 2(R - 18)
(R + 6) - 18 = R - 12
R - 12 = 2(R - 18)
R - 12 = 2R - 36
24 = R
What is exponential growth?
When a certain quality increases a specific amount each specific unit of time.
(Ex., doubles every year)
What is exponential decay/decline?
When a certain quality decreases a specific amount each specific unit of time
(Ex., halves every two years)
Every 12 years, a certain village has been losing 10% of its population. If 1000 people lived in the village in 1960, What was its population in 1996?
1996 - 1960 = 36
1000 x 0.9(to the power of 36/12)
1000 x 0.9³
1000 x 0.729
729
A certain bacteria triples every twenty minutes. How long will it take for 1 million bacteria to reach 27 million?
27 = 1 x 3(to the power of x/20) 3³ = 3(to the power of x/20)
3 = x/20 60 = x
Temperature drops in half for every mile rise. The surface temperature is 96 degrees. What is the temperature six miles above?
1 mile = 0.5
96 x 1/2⁶
96 ➗ 2/1⁶
96 ➗ 64
1.5
Time =
Distance / Speed
Speed =
Distance / Time
Distance =
Speed x Time
A train travels 210 miles from Boston to NYC at 70 mph. Afterwards, it then travels 90 miles from NYC to Philadelphia at 45 mph. What was the average speed for the journey?
(Boston) 210/70 = 21/7 = 3 hours
(Philadelphia) 90/45 = 2 hours
3(70) + 2(45) / 5
210 + 90 / 5
300/5
60
The total distance of a trip is 1680 miles. The plane flies the first 900 miles in 3 hours. What speed must it fly the remaining miles to average 280 mph for the entire journey?
280 = 1680/x 280x = 1680 x = 6
1680 - 900 = 780
780/(6-3)
780/3
260 mph
A freight truck travels to Seattle to Vancouver back to Seattle in four hours. The first trip had an average speed of 60 mph. The second trip had an average speed of 45 mph. How far is Seattle from Vancouver?
S => V => S in 4 hours
S => V is X miles
60 = 5 x 3 x 4 45 = 5 x 3 x 3
4 = x/60 + x/45 4 = 3x/180 + 4x/180 4 = 7x/180 720 = 7x 102.85 = x
A student can walk from his house to school in M minutes at a rate of 2 mph. In an additional 15 minutes, he can jog to school and back. What is his jogging speed?
d = 2 x M/60 d = 1 x M/30 d = M/30
s = 2(M/30) / (M + 15)/60 s = 2M/30 x 60/(M + 15) s = 2M/1 x 2/(M + 15) s = 4M/(M + 15)
On a 550 mile road trip, Jessica drives the first 375 miles in 7.5 hours. At what speed must she drive the remaining distance to average 55 mph for the entire trip
Whole: 550 mi, 55 mph, *10 hrs (T=D/S, 550/50)
First: 375 mi, *50 mph, 7.5 hrs (S=D/T, 375/7.5)
Second: *175 mi, ? mph, *2.5 hrs
S = D/T = 175/2.5 = 70 mph
I spend 1 3/4 hours every day in traffic going from my office and coming back home along the same route. I average 64 driving to the office and 48 mph driving home. How far is my office from my home?
Home to Office = x/64
Office to Home = x/48
1 3/4 = x/64 + x/48 7/4 = x/64 + x/48 7 = 4x/64 + 4x/48 7 = x/16 + x/12 7 = 3x/48 + 4x/48 7 = 7x/48 7(48) = 7x 48 = x
At 2:30, two trains start towards each other on the same track 230 miles apart. One train averages 55 mph, the other 60 mph. At what time will they hit?
T = D/S
D = 230 mi
S = 55 + 60 = 115
230/115 = 2
2:30 + 2 = 4:30
At 10:00, Jason leaves home at a speed of 50 mph. At 10:30, Sam leaves in the opposite direction at a speed of 45 mph. How far away will they be from each other at 1:30?
D = T x S
10: 00 to 1:30 = 3.5 hours
10: 30 to 1:30 = 3 hours
3.5(50) = 175 3(45) = 135
175 + 135 = 310
A fighter jet is flying at 700 mph. A missile is fired from 300 miles away traveling at 1900 mph. How long will it take for the missile to strike the jet?
T = D/S
S = 1900 - 700 = 1200 mph D = 300 miles
T = 300/1200 = 3/12 = 1/4
15 minutes.
A plane leaves the airport at 1:00 at a speed of 300 mph. A second plane leaves the same airport 15 minutes later at a speed of 450 mph. At what time will the second plane meet the first?
D = S x T D = 300 x 1/4 = 75
T = D/S
S = 450-300 = 150
T = 75/150 = 1/2 or 30 minutes
1:45
An ant can eat a sugar crystal in 15 minutes. A cockroach can eat it thrice as fast. If they eat it together for 3 minutes, what fraction of the sugar crystal will remain to be eaten?
1/15 + 3/15 = 4/15
3(4/15) = 4/5
1 - 4/5 = 1/5
Pump X takes 9 hours to empty a pool. Pump Y takes 6 hours. Both pumps are used to drain the pool but Pump Y overheats and shuts down after 3 hours. How many more hours will Pump X need to finish emptying the pool?
1/9 + 1/6 = 2/18 + 3/18 = 5/18
3(5/18) = 5/6
1 - 5/6 = 1/6
x/9 = 1/6 x = 9/6 = 3/2 = 1 1/2
1.5 hours
Victor can assemble a sofa alone in H hours. If Victor and Sundeep can finish the sofa together in 2 hours, how long would it take Sundeep to assemble the sofa alone?
1/h + 1/x = x/hx + h/hx = (x + h)/hx
2(x + h)/hx = 1 (2x + 2h)/hx =1 2x + 2h = hx 2h = hx - 2x 2h = x(h - 2) 2h/(h - 2) = x
Sophia can wash her car in H hours. If she and her husband work together, it takes them M minutes. How long would it take her husband to wash the car alone?
1/60h + 1/x = x/60hx + 60h/60hx = (x + 60h)/60hx
m(x + 60h)/60hx = 1 mx + 60hm = 60hx 60hm = 60hx - mx 60hm = x(60h - m) 60hm/(60h - m) = x
What is the formula for AMOUNT of work done by multiple people at the same rate?
(Amount of Work) = (# of Workers) x (Time Needed)
What is the formula for TIME NEEDED for work done by multiple people at the same rate?
(Time Needed) = (Amount of Work) / (# of Workers)
What is the formula for NUMBER OF WORKERS NEEDED for work done by multiple people at the same rate?
(# of Workers) = (Amount of Work) / (Time Needed)
It takes 4 painters 5 days to paint a house. How many more painters would be needed to paint the house in 3 days less time?
Work = 4 x 5 = 20
20 = (4 + x)(5 - 3) 20 = (4 + x)(2) 20 = 8 + 2x 12 = 2x 6 = x
It takes 2 masons 15 hours to build a wall. How much less time would it take 5 masons?
W = 2 x 15 = 30
30 = 5x 6 = x
15 - 6 = 9
Two cows can consume a sack of fodder in 3 days. How many fewer days would it take 6 cows to consume the same bag of fodder?
W = 2 x 3 = 6
6 = 6(3 - x) 6 = 18 - 6x 6x = 18 - 6 6x = 12 x = 2
Put into an equation…
5 more than x
x + 5
