Students often complain that word problems are the most difficult part of algebra. But there are ways to make them easier to figure out. I see them as four main groups with some overlap Translations, Structures, Assumed Knowledge, and Formulas. Within each are several types. One way of making word problems much easier is to identify which type they are. Then you can have an idea of where you are headed as you develop the right equation.
Once you have determined what type of word problem you have, then you can begin to think about how to set it up in terms of translating, developing a chart or drawing, or remembering a formula. These set ups can guide you to what information or numbers to look for in the word problem, and where to place them to build your equation. [4]
This page will highlight the following types of word problems.
Structure – Chart/Drawing/Formulas – DRT
Traveling is usually going on in these problems, with phrases involving distance, time, and rate/speed/mph.
Speed of the Current and the Paddler
Literal Translation / Math Language / Numbers
These problems use math language to indicate what is going to happen with numbers. In short forms, they are like translating a foreign language. In the longer forms, situations will involve numbers of things (years, sales, people) that will usually form a sum (total), but may form a difference, product or quotient.
Structure – Chart/Drawing/Formulas – Mixture
Mixing liquids (% solutions) to a goal middle rate is most common, but these problems can also involve mixing things that have different prices (nuts, candy, flowers, etc.) to a goal middle price (rate).
Formula – Given
These word problems may seem daunting, but are actually some of the easiest. A formula is given. Simply read the word problem for what each variable stands for and which ones are given. Plug them into the formula and solve for the remaining unknown variable.
Distance = Rate * Time
Speed of the Current
Rick paddled up the river, spent the night camping, and and then paddled back. He spent 10 hours paddling and the campground was 24 miles away. If Rick kayaked at a speed of 5 mph, what was the speed (rate) of the current? [1]
Use CUBES to solve the problem.
- Circle
- 10 hours paddling
- Campground 24 miles away
- Speed was 5 mph
- Underline
- What was the speed of the current given that Rick’s speed was 5 mph?
- Box
- Rick kayaked at a speed of 5 mph
- Evaluate
- Spent the night camping (Eliminate any information you do not need.)
- Solve and Check
Let the speed of current be r mph
When he paddled up the river (toward the source of a river), he decreases his speed (going against the current), so the rate is 5-r
When he paddled down the river, he increases his speed (going with the current), so the rate is 5+r
Use the distance formula to complete the table below.
| Type | Rate | Time | distance = rate * time |
|---|---|---|---|
| Up River | 5 – r | 24/(5 – r) | 24 miles |
| Down River | 5 + r | 24/(5 + r) | 24 miles |
As the total time for up and back is 10 hours. So, we solve for total distance/rate = time.
24/(5-r) + 24/(5 + r) = 10
24(5 + r) + 24(5-r) = 10(5 + r)(5 – r)
120 – 24r + 120 + 24r = 250 – 10r2
240 = 250 – 10r2
10r2 = 10
r2 = 1
r = ±1
Because we are solving for the speed of the current [3], we will discard the negative value. (We are looking for the principle square root.) Thus, the speed of current is 1 mph.
Speed of the Current and the Paddler
Mr. R went kayaking in a river. Paddling at a constant rate against the current for 20 km from Town A to Town B took him 4 hours. The return trip with the current took 2.5 hours. Rate of current (c) and paddler (r)? [2]
Note that in the problem above we solved a quadratic function ( f(r) = r2 – 1 ) to get the answer. In this question, we solved two simultaneous linear equations. (See graph on Desmos.)
distance = rate * time
20 = (r – c) * 4
20 = (r + c) * 2.5
Simplify each equation.
5 = r – c
8 = r + c
Add the equations to eliminate the variable c.
13 = 2r
r = 13/2 = 6.5 kph, i.e., the rate of the paddler.
Use one of the simplified distance equations to solve for c, the rate of the current.
8 = 6.5 + c
8 – 6.5 = c
c = 1.5 kph, i.e., the rate of the current.
How Old Is Each Now?
Emma is 8 years older that her brother Landon. Three (3) years ago, she was 3 times older that her brother. How old is each now?
Given the information above, we know the following:
Emma = Landon + 8
E is Emma’s age and L is Landon’s age
∴ E = L + 8
What happened three (3) years ago?
E – 3 = 3(L – 3), set E to L + 8 to get one variable to solve for.
(L + 8) – 3 = 3(L – 3)
L + 5 = 3L – 9 (See graph on Desmos.)
14 = 2L
7 = L (Landon’s age) ∴ E = 15 (Emma’s age)
Three years ago, Emma was 12 and Landon was 4, so Emma was 3 times older than her brother.
A Very Berry Snoogle®
Kiedrowski’s Bakery wants to make 30 pounds of berry mix that costs $3 per pound to use in their Snoogle® mix. They are using blueberries (x) that cost $2 per pound and blackberries (y) that cost $3.50 per pound. How many pounds of blackberries should be used in this mixture?
x + y = 30
2x + 3.5y = 90 (e.g., $2/lb. * x lbs. + $3.5/lb. * y lbs. = 30 lbs. * $3/lb.)
x = 30 – y, substitute into 2x + 3.5y = 90.
2(30 – y) + 3.5y = 90, solve for y.
60 – 2y + 3.5y = 90
1.5y = 30
y = 20
x = 30 – 20 = 10
∴ Kiedrowski’s Bakery needs 10 lbs. of blueberries and 20 lbs. of blackberries. (See graph on Desmos.)
Money Word Problem
Formula for Simple Interest
i = prt
i represents the interest earned.
p represents the principal which is the number of dollars invested.
r represents the interest rate per year.
t represents the time the money is invested which is generally stated in years or fractions of a year.
Formula for Amount
A = p + i
A represents what your investment is worth if you consider the total amount of the original investment (p) and the interest earned (i)
Example
James needs interest income of $5,000. How much money must he invest for one year at 7%? (Give your answer to the nearest dollar.)
Solution:
5,000 = p(0.07)(1)
p = 71,428.57
He must invest $71,429
References
[1] “Q. 222 Rick Paddled Up The River, Spent… [FREE SOLUTION] | StudySmarter”. 2023. StudySmarter US. https://www.studysmarter.us/textbooks/math/intermediate-algebra-oer-2017/quadratic-equations-and-functions/q-222-rick-paddled-up-the-river-spent-the-night-camping-and-/.
[2] “Mr. R Went Kayaking In A River. Paddling At A Constant Rate Against The Current For 20 Km From Town A To Town B Took Him 4 Hours. The Return Trip With The Curre | Wyzant Ask An Expert”. 2023. wyzant.com. https://www.wyzant.com/resources/answers/174849/mr_r_went_kayaking_in_a_river_paddling_at_a_constant_rate_against_the_current_for_20_km_from_town_a_to_town_b_took_him_4_hours_the_return_trip_with_the_curre.
[3] Can a scalar be negative?
- Scalars are quantities that are fully described by a magnitude (or numerical value) alone.
- Generally, scalars in the physical world are positive, but some scalars like electric charge and temperature have a negative value.
- In conclusion, we can say that scalar quantity can be negative.
[4] “Word Problem Types”. 2023. faculty.mtsac.edu. https://faculty.mtsac.edu/ctunstall/dsps_33/dsps33_maths/Word%20Problem%20Types.pdf.
Additional Reading
“3: Math Models – Mathematics LibreTexts”. 2023. Mathematics LibreTexts. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_098_Elementary_Algebra/3%3A_Math_Models.
“3.2: Use A Problem-Solving Strategy”. 2020. Mathematics LibreTexts. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_098_Elementary_Algebra/3%3A_Math_Models/3.2%3A_Use_a_Problem-Solving_Strategy.
“Math Word Problems (Video Lessons, Examples And Step-By-Step Solutions)”. 2023. onlinemathlearning.com. https://www.onlinemathlearning.com/math-word-problems.html.
These lessons will illustrate how word problems can be solved using block diagrams. Students who have not yet learned algebra can use the block diagrams or tape diagrams to help them visualize the problems in terms of the information given and the data to be found. This allows the student to decide which operators to use: Addition, Subtraction, Multiplication or Division. Block diagrams or tape diagrams are used in Singapore Math and Common Core Math.
Mathematical Mysteries 