Word Problems Flashcards
What do the following words translate to, in algebraic symbols?
a) is
b) of
c) per
d) percent
e) what
f) half as many
a) =
b) * (times)
c) /
d) /100
e) w (variable we are trying to find)
f) *1/2
Translate the following into Algebraic Expressions or Equations:
a) There are 5 less B than A
b) x is 5 times as large as 2 less than y
c) 3x percent of y
a) B = A - 5
b) x = 5(y-2)
c) 3xy/100 (percent –> /100, of –> *)
Translate into equations:
a) James is twice as old as Mary was 2 years ago
b) If the Bus were 3 feet longer, it would be 4 times as long as the car
c) 20% more than A is 500 less than B
a) J = 2(M-2)
b) B + 3 = 4C
c) 1.2A = B - 500
Translate:
In 5 years, Adam will be twice as old as Bob.
A + 5 = 2(B + 5)
The trick here is to compare the 2 ages in the future!
A common mistake is A + 5 = 2B
Translate into equations and solve:
200 pounds of apples are divided into small and large bags, of 5 and 10 pounds respectively.
There are 4 more small bags than large bags.
How many large bags are there?
Define variables: S = small bags, L = large bags
1) 200 = 5S + 10L
2) S = L + 4
Substitute 2 into 1, so we only have one variable, L
200 = 5(L + 4) + 10L
200 = 5L + 20 + 10L
180 = 15L
L = 12
Small boxes are $3, large boxes are $5.
If John buys 10 boxes for $36, how many are small?
Solve by writing the equations, or use your own intuition and logic.
S = small, L = large
(1) S + L = 10
(2) 3S + 5L = 36
The question asks for S. So, solve (1) for L, so we can then substitute it into (2) and solve for S.
L = 10 -S
3S + 5(10 - S) = 36
- 2S + 50 = 36
- 2S = 14 –> S = 7
Intuition method: If they were all small, it would cost 10*3 = $30
Larges are $2 more than smalls, so if we swap 3 smalls for larges, it would cost $36.
What is the equation for Distance or Work?
RT = D
RT = W
R = Rate (miles per hour, jobs per day, etc)
T = time
A bus travels 720 miles at 20 miles per hour. How long does this take?
RT = D
T = D / R
T = 720 / 20 = 36 hours
Quickest way to do the division is drop the 0 and divide by 2. (divide by 10 and then 2)
John travels 480 feet in 2 minutes. What is his speed, in feet per second?
Notice the Units! We need to convert minutes to seconds.
2 minutes * 60 seconds/min = 120 seconds
RT = D
R = D/T
R = 480 / 120 = 4 ft/s
Joe jogs a 26 mile course at 4 miles per hour.
If Mary jogs the course in 90 fewer minutes, how fast did she jog?
First, find time for Joe:
Joe: T = D/R = 26 / 4 = 6.5
Mary takes 90 fewer minutes –> convert to hours –> 1.5 fewer hours –> 5 hours
Mary: R = D/T = 26 / 5 = 5.2 miles per hour
Relative Rates:
A and B are 20 miles apart at 1pm
A walks towards B at 2mph.
B walks towards A at 3mph.
What time will they reach each other?
We can add their rates together, since they are going towards each other.
Combined Rate = 2+3 = 5
T = D/R = 20/5= 4 hours
1pm + 4 hours = 5pm
A starts 35 miles west of B.
They both head east at the same time.
A travels 30mph
B travels 25mph
After how many hours will A and B meet?
Relative Rates:
They are both heading the same direction, so A is catching up to B at the difference between their speeds:
Closing Speed = 30 - 25 = 5mph
T = D/R = 35 / 5 = 7 hours
Average Rate:
Joe walks to work at 3mph, and walks home at 2mph.
What is his average speed for the entire trip?
Intuitively speaking, will the average speed be closer to 2mph or 3mph?
Average Rate: Find the Total Time!
You can’t just average the 2 rates! (the reason is Joe spends more time walking at the slower rate)
You must find the total time and total distance, and then divide:
Pick a number for the distance each way, such as 6 miles:
Going: T = 6/3 = 2 hours
Return: T = 6/2 = 3 hours
Total T = 5 hours, Total D = 12 miles
R = 12/5 = 2.4 mph
Intuitively, it makes sense that the speed is closer to 2mph because Joe spent more time walking at that speed, and average speed equals total_distance/total_time
Joe can sand 2/5 of the table in 6 hours.
How long will it take him to sand the whole table?
6 workers can complete 1/2 of a job in 5 days.
a) What is the rate of 1 worker?
b) How many days would it take for 2 workers to complete a full job?
a) RT = W
Let R = Rate of 1 worker
There are 6 workers, so 6R * 5 =1/2
30R = 1/2
R = 1/60 jobs/day
b) If 1 worker has a rate of 1/60, 2 workers have a rate of 2/60 = 1/30 jobs/day
Take the reciprocal to find time: 30 days
Translate into an equation:
Adam takes 3 hours to do a job, Bob takes x hours to do a job, and together they take y hours to do a job.
We can’t add the Hours together.
We have to turn them into Rates–> then, we can add them
1/3 + 1/x = 1/y
Worker A takes 4 hours to do a job.
Worker B takes 6 hours to do a job.
Working together, how long would it take them to complete 2 jobs?
Add the Rates:
A: R = 1/4 jobs/hour
B: R = 1/6 jobs/hour
Together: R = 1/4 + 1/6 = 3/12 + 2/12 = 5/12 jobs/hour
We need to find T for 2 jobs:
T = W / R = 2 / (5/12) = 24 / 5 = 4.8 hours
If A is a subset of B, what does that mean?
All the elements of the set A are in set B.
Example 1: if A is the set {1,8} and B is the set {1, 3, 5, 8}, A is a subset of B
Example 2: if A represents all male employees, and B represents all employees, A is a subset of B.
Data sufficiency:
Of 20 houses built last year in city Y, how many were occupied at the end of the year?
(1) Of all the houses in city Y, 50 percent were occupied at the end of last year
(2) A total of 100 houses in city Y were occupied at the end of last year
E- Not sufficient
Read very carefully! The key words are “last year” in the stem. We need to find number in the orange circle, which is a subset of the 20 built last year. The clues talk about ALL houses, not just the ones built last year, so they don’t help us.
What is the definition of “Median”?
The “middle” value, when numbers are listed from least to greatest:
If there are an odd number of terms, it will be the center term:
1, 3, 6, 8, 10 –> Median is 6
If there are an even number of terms, it will be the average of the two middle terms:
1, 3, 5, 8 –> the Median is between 3 and 5 – > it is the midpoint between them, or average, which is 4.
If there are 15 terms listed from least to greatest, which value will be the median?
Example: With 5 terms, the 3rd is the median
The 8th term.
Formula for odd number of terms: (n+1) / 2
What is the median of the following terms?
2
1
5
8
3.5
First, list from least to greatest:
1, 2, 5, 8
Even number of terms, so median is average of 2 and 5
(2+5) / 2 = 3.5
What is the median of the following:
x+11
x+3
x
x+15
x+4
x+4
List from least to greatest:
x, x+3, x+4, x+11, x+15
The middle is x+4
What is the formula for “average” (also called “mean”)?
What is the formula for “Sum”?
Average = Sum / number of terms
A = S/n
or, S = An
example: the Average of {2, 6, 7} = (2+6+7)/3
= 15/3 = 5
If the average of 8 scores is 21, what is the total of all the scores?
168
S = An = 8*21 = 168
If k = 4, what is the average of:
k
k+2
k-3
3k+1
Sum = k + (k+2) + (k-3) + (3k+1) = 6k = 6*4 = 24
A = S/N = 24/4 = 6
If the average of the set {1, 4, 3, 7, 8, x} is 5, what is x?
7
Sum = Average * number of terms (S=An)
S = An = 5*6 = 30
1+4+3+7+8+x = 30
23+x = 30
x = 7
For the given set, how much higher is the mean (average) than the median?
{3, 4, 4, 5, 11, 15)
2.5
Mean: A = S/n = 42 / 6 = 7
Median for even # of terms = average of middle 2 values
= average (midpoint) of 4 and 5 = 4.5
7 - 4.5 = 2.5
John’s 10th sale, $1200, raises his average sale to $300. What was his average sale before the 10th sale?
$200
After 10th sale: S = An = 10*300 = 3000
Before 10th sale, there were 9 sales. n = 9
The sum was 3000 - 1200 = 1800
A = S / n = 1800 / 9 = $200
Which set has a larger standard deviation?
A: {2, 4, 6}
B: {3, 4, 5}
A
Both sets have the same average, 4, but the numbers in A are more spread out from the average.
If each term in a set is increased by 5, does the set’s standard deviation increase, decrease, or stay the same?
Stay the same
If the set were plotted on a number line, it would still be spread out in the same pattern, with the same amount of spread between the terms, but it would just slide 5 units to the right.
If each term in a set is multiplied by 1/2, does the set’s standard deviation increase, decrease, or stay the same? (assume that the set includes different numbers)
Decrease
As seen below, the data will get closer together. Each point’s distance from the mean will be cut in half.
If the set’s numbers were all the same, the standard deviation wouldn’t change, because it would be 0 for both.
What does “Standard Deviation” mean?
(you don’t need the the formula, just the concept)
Standard Deviation measures the spread or variation of the data in a set.
It indicates how far the data points are from the average (mean)
What does a standard deviation of 0 signify?
A standard deviation of 0 means all numbers in the set are equal.
How many integers are there between 5 and 11, inclusive?
7
A common mistake is to subtract and get 6
Inclusive means it includes the endpoints, 5 and 11.
5, 6, 7, 8, 9, 10, 11
Our formula is (Last - First) / Increment + 1
“Increment” means the difference between the numbers in an Evenly Spaced Set
ex. {5, 8, 11, 14} –> Increment = 3
What is the formula for the sum of consecutive integers between 1 and n, inclusive?
For the sum of integers from 1 to n,
Sum = Average * Number of Terms simplifies to:
Using the formula, what is the sum of the integers between 1 and 10, inclusive?
How many multiples of 5 are there between 74 and 128?
What is the formula?
11
(Last - First) / Increment + 1
First is 75. Last is 125. Increment is 5.
(125 - 75) / 5 + 1 = 11
For an evenly spaced set, such as {3, 5, 7, 9} what is true about the mean and median?
They are equal.
An evenly spaced set means the difference is the same between each term.
What is the formula for the mean and median of an evenly spaced set?
What is the mean and median of {6, 12, 18, 24, 30, 36} ?
(First + Last) / 2
(6+36) / 2 = 21
For an evenly spaced set, the mean and median are the same
What is the formula for the sum of consecutive integers?
For an evenly spaced set:
Formula: Sum = Average * Number of Terms
Average = (First + Last) / 2
Number of Terms = ((Last - First) / Increment) + 1
For “consecutive integers”, the increment is just “1”
Note: if it said “EVEN consecutive integers”, the increment would be “2”, since the numbers are 2 apart (2, 4, 6…)
What is the sum of all the ODD integers between 10 and 50, inclusive?
600
For an evenly spaced set:
Formula: Sum = Average * Number of Terms
Average = (First + Last) / 2 = (11 + 49) / 2 = 30
(Note: “First” and “Last” are 11 and 49 because the question asks for ODD integers; be careful not to automatically use 10 and 50!)
Number of Terms = ((Last - First) / Increment) + 1 = ((49 - 11) / 2) + 1 = 19 + 1 = 20
(Note: Increment = 2 because it says “all the ODD integers”; “11, 13, 15…”)
Sum = Average * Number of Terms = 30 * 20 = 600
