C. Polynomial and Rational Functions · Rational Functions and Equations

Applications: Work, Travel, Average Cost and Concentration

C3.7About 75 minutes

Learning goals

I can model work-rate problems with rates such as 1/40 of a job per minute and solve the rational equation.
I can model travel problems with time = distance ÷ speed, including currents and winds.
I can use average-cost and concentration models and interpret their asymptotes.
I can reject solutions that do not make sense in the context.

Words to know

rate of work
The fraction of a job done per unit of time. Someone who finishes a job in t hours works at a rate of 1/t of the job per hour.

average cost
Total cost divided by the number of items: C(n) = (fixed cost + variable cost × n)/n.

concentration
The fraction (or percentage) of a mixture that is one ingredient: amount of ingredient ÷ total amount.

fixed cost
A cost that does not change with the number of items, such as buying a machine.

Where rational equations come from

Rational equations appear whenever something is divided by an unknown: a time is distance ÷ speed, a rate is 1 ÷ time, an average is a total ÷ a count, and a concentration is an amount ÷ a volume. Each type of problem has its own pattern. Learn the pattern, and the algebra is the same as in the last two lessons.

Applications of polynomial models (for example, problems that use the remainder or factor theorem) are covered in the unit Polynomial Functions and Equations. This lesson focuses on rational models.

Pattern 1: work problems

If a person can do a job alone in t minutes, they do 1/t of the job each minute. When people work together, their rates add:

(rate of A) + (rate of B) = (rate together), so 1/a + 1/b = 1/t,

where a and b are the times alone and t is the time together.

Worked example

Example 1: Shovelling the driveway

After a big snowfall, Amira can clear the driveway alone in 40 minutes. Her dad needs 60 minutes. How long do they take together?

Rates: Amira 1/40 of the driveway per minute; Dad 1/60 per minute.
Together: 1/40 + 1/60 = 1/t.
Multiply by the LCD 120t: 3t + 2t = 120, so 5t = 120 and t = 24.
Check: in 24 min Amira does 24/40 = 3/5 and Dad does 24/60 = 2/5; together 1 whole driveway ✓.

24 minutes. (Sensible: faster than either one alone.)

In 24 minutes Amira clears 3/5 of the driveway and her dad clears 2/5. The rates add to one whole job.

Worked example

Example 2: An unknown time

A new snowblower clears a parking lot 1 h faster than an old one. Working together they take 1.2 h. How long does each take alone?

Let x be the old machine's time (h). The new one takes x − 1.
1/x + 1/(x − 1) = 1/1.2. Multiply by 1.2x(x − 1): 1.2(x − 1) + 1.2x = x(x − 1).
2.4x − 1.2 = x² − x, so x² − 3.4x + 1.2 = 0. Multiply by 5: 5x² − 17x + 6 = 0.
(5x − 2)(x − 3) = 0, so x = 0.4 or x = 3.
Reject x = 0.4: the new machine's time would be 0.4 − 1 < 0, which is impossible.
Check x = 3: 1/3 + 1/2 = 5/6, and 1/1.2 = 5/6 ✓.

Old: 3 h; new: 2 h.

Pattern 2: travel problems

Use time = distance ÷ speed. With a current (or wind) of speed c, a boat with still-water speed v goes v + c downstream (with the current) and v − c upstream (against it).

Worked example

Example 3: Paddling against the current

A canoeist paddles at 5 km/h in still water. She paddles 8 km upstream and 8 km back in 3 h 20 min. Find the speed of the current.

Total time: 3 h 20 min = 3 1/3 h = 10/3 h.
8/(5 − c) + 8/(5 + c) = 10/3, with 0 ≤ c < 5.
Left side over a common denominator: [8(5 + c) + 8(5 − c)]/(25 − c²) = 80/(25 − c²).
So 80/(25 − c²) = 10/3, which gives 25 − c² = 24, so c² = 1 and c = 1 (c ≥ 0).
Check: 8/4 + 8/6 = 2 + 4/3 = 10/3 ✓.

The current is 1 km/h.

Upstream, the current slows the canoe to 5 − c; downstream it speeds it up to 5 + c. Each leg's time is distance ÷ speed.

Pattern 3: average cost

If making n items costs a fixed amount F plus v dollars per item, the average cost per item is C(n) = (F + vn)/n. This is a linear-over-linear rational function. Its horizontal asymptote C = v means that for very large orders the fixed cost is spread so thin that each item costs almost exactly v.

Worked example

Example 4: Maple-leaf pins

A school club pays $120 for a pin-making press, plus $1.50 in materials per pin. How many pins must it make for the average cost to be $2.30 per pin?

C(n) = (120 + 1.5n)/n.
(120 + 1.5n)/n = 2.3. Multiply by n (n > 0): 120 + 1.5n = 2.3n.
120 = 0.8n, so n = 150.

150 pins. The average cost falls toward the asymptote $1.50 as n grows, but never reaches it.

The average cost starts high (the $120 press is shared by few pins) and falls toward the horizontal asymptote C = 1.5. It reaches $2.30 when n = 150.

Pattern 4: concentration

Concentration = (amount of ingredient) ÷ (total amount). Adding x litres of something changes both the top and the bottom, which is what makes the equation rational.

Worked example

Example 5: Stronger juice

A 10 L jug of punch contains 2 L of juice concentrate (20%). How much concentrate must be added to make the punch 36% concentrate?

After adding x L: concentration = (2 + x)/(10 + x).
(2 + x)/(10 + x) = 0.36, so 2 + x = 3.6 + 0.36x.
0.64x = 1.6, so x = 2.5.
Check: 4.5/12.5 = 0.36 ✓.

2.5 L of concentrate.

Just for fun

Every work-rate problem quietly assumes that two people shovelling the same driveway never stop for a snowball fight. It may be the least realistic assumption in all of mathematics.

Hands-on activity

The card-sorting race

You need: a shuffled deck of 52 playing cards, a timer (a phone works), a table, and a partner.

You alone: time yourself sorting the deck into four piles by suit. Record your time a (in seconds).
Partner alone: shuffle and time your partner. Record b.
Predict: use 1/a + 1/b = 1/t to predict the time t if you split the deck and work at the same time, each sorting into the same four piles.
Test: shuffle, split the deck into two halves of 26, and sort together. Record the real time.
Compare. Was the real time longer or shorter than predicted? Why might that be?

What to look for: for example, a = 60 s and b = 90 s predicts t = 36 s. Real teams often take a little longer than predicted (you bump hands or wait for each other), or finish at different moments because each person has exactly half the deck. The model assumes both people work at steady rates on one shared job until it is done.

Your turn

Pipe A fills a tank in 6 h and pipe B fills it in 3 h. How long do they take together?

Two people paint a fence together in 4 h. One of them alone takes 6 h. How long would the other take alone?

A train travels 300 km. If its speed were 10 km/h faster, the trip would take 1 h less. Find its speed.

A shop pays $200 to set up a hoodie design, plus $15 per hoodie. How many hoodies give an average cost of $20 each?

A 12 L punch bowl contains 3 L of juice (25%). How much juice must be added to make it 40% juice?

A number plus its reciprocal is 10/3. Find the number.

More practice: warm-up

A job takes 5 h. What fraction of it is done each hour?

A machine finishes a task in 20 min. Write its rate.

Write an expression for the time to travel 150 km at v km/h.

Write the average cost per item when the fixed cost is $50 and each item costs $2.

A jug holds 8 L of drink, of which 2 L is juice. What is the juice concentration?

More practice: on your own

Hose A fills a backyard pool in 10 h and hose B in 15 h. How long with both?

A kayaker paddles at 6 km/h in still water. She goes 9 km downstream and 9 km back in 4 h. Find the speed of the current.

A small plane flies 600 km against a 40 km/h headwind, then 600 km back with the same wind behind it. The round trip takes 6 h 15 min. Find the plane's speed in still air.

An older printer takes 3 h longer than a newer one to print a yearbook order. Together they take 2 h. How long does each take alone?

For the pin club (C(n) = (120 + 1.5n)/n), how many pins give an average cost of $1.75?

Challenge

An inlet pipe fills a tank in 4 h, and a drain empties the full tank in 6 h. If both are open, how long does the empty tank take to fill?

300 g of a maple syrup mixture is 20% sugar by mass. How much water must be boiled off to make it 30% sugar? (Boiling syrup is much hotter than boiling water and can cause serious burns, so it is a job for an adult.)

A cyclist rides 30 km to a town, then rides back 5 km/h slower. The return trip takes 30 min longer. Find both speeds.

Check your understanding

Quiz

Amira clears a driveway in 40 min and her dad in 60 min. How long do they take together?

One person can do a job in 3 h. With a second person, it takes 2 h. If the second person alone takes x hours, which equation models this?

A canoe moves at 5 km/h in still water and the current is c km/h. What is its speed going upstream?

What does the horizontal asymptote C = 1.5 of C(n) = (120 + 1.5n)/n mean?

A 10 L jug holds 2 L of concentrate. After x L more concentrate is added, what is the concentration?

In the snowblower problem the quadratic gave x = 3 or x = 0.4 for the old machine's time. Why is 0.4 rejected?

For parents and teachers

Big idea (C3.7): many real situations divide by an unknown (time = distance ÷ speed, rate = 1 ÷ time, average = total ÷ count, concentration = part ÷ whole), and so lead to rational equations. Solve them as before, then check that the answer makes sense in context.

How to help: The card-sorting race is a fun way to test the work-rate model. Ask your teen to explain why the predicted time together must be less than either time alone. In the syrup question, any boiling is a job for an adult.

Common difficulties:

Adding times instead of rates (40 + 60 is not the time together).
Writing v − c and v + c the wrong way round for upstream and downstream.
Forgetting that adding an ingredient changes the total volume too.
Keeping a negative or impossible root without checking the context.