Analytic Geometry · Analytic Geometry

Rate, Mixture and Investment Problems

1.2About 70 minutes

Learning goals

I can model current and wind problems with a linear system.
I can model mixture problems by adding totals and adding amounts of an ingredient.
I can model simple-interest investment problems with a linear system.
I can check that my answer is reasonable in the situation.

Words to know

rate
A comparison of two quantities with different units, such as km/h or $/kg.

concentration
The percent of a mixture that is a certain ingredient.

simple interest
Interest calculated only on the original principal: I = Prt.

principal
The amount of money invested or borrowed.

GIC
Guaranteed Investment Certificate: a Canadian savings investment that pays interest, usually at a fixed rate, for a fixed term.

headwind / tailwind
Wind blowing against / in the same direction as a moving plane.

Rates, mixtures and money

Some of the most useful linear systems come from three kinds of problems:

Rate problems: a canoe on a river, a plane in the wind. Distance = speed × time.
Mixture problems: blending two solutions, two kinds of nuts, two juice drinks. Amount of ingredient = percent × total amount.
Investment problems: money split between two accounts with different interest rates. Interest = principal × rate × time.

In each type, one equation usually adds up the totals and the other adds up the ingredient (or speeds, or interest). A table helps you organize the information before you write equations.

Rate problems: currents and winds

When a canoe goes downstream, the current helps it, so its speed over the ground is the paddling speed plus the current. Going upstream, the current works against it, so its speed is paddling speed minus the current. Planes work the same way with a tailwind (helping) and a headwind (against).

The current adds to the canoe's speed going downstream and subtracts from it going upstream. The shaded band is the river.

Worked example

Example 1: Canoe on the river

A canoeist paddles 12 km downstream in 1.5 h. The return trip upstream takes 3 h. Find her paddling speed in still water and the speed of the current.

Define: let c be her paddling speed in still water and w the speed of the current, both in km/h.
Find each ground speed using speed = distance ÷ time. Downstream: 12 ÷ 1.5 = 8 km/h. Upstream: 12 ÷ 3 = 4 km/h.
Equations: c + w = 8 and c − w = 4.
Solve by elimination (add): 2c = 12, c = 6. Then 6 + w = 8, w = 2.
Check in the words: downstream 6 + 2 = 8 km/h, so 12 km takes 12 ÷ 8 = 1.5 h ✓. Upstream 6 − 2 = 4 km/h, so 12 km takes 3 h ✓.

She paddles at 6 km/h in still water and the current is 2 km/h.

Worked example

Example 2: A plane and the wind

A plane flies 1800 km with a tailwind in 3 h. Against the same wind, the return flight takes 4 h. Find the plane's airspeed (its speed in still air) and the wind speed.

Let p be the airspeed and w the wind speed in km/h.
With the wind: 1800 ÷ 3 = 600 km/h, so p + w = 600. Against: 1800 ÷ 4 = 450 km/h, so p − w = 450.
Add: 2p = 1050, p = 525. Then w = 600 − 525 = 75.
Check: 525 − 75 = 450 ✓; 450 × 4 = 1800 ✓.

Airspeed 525 km/h, wind 75 km/h.

Mixture problems

A 20% acid solution means that 20% of the solution is acid. So 20 L of it contains 0.20 × 20 = 4 L of acid. When you mix two solutions:

the total amounts add (litres + litres), and
the amounts of the ingredient add (litres of acid + litres of acid).

The percents do not add. Mixing a 20% solution and a 50% solution can never give a 70% solution; the result is always somewhere between 20% and 50%.

Mixing 20 L of a 20% solution with 10 L of a 50% solution. The volumes add and the amounts of acid add, but the percents do not: the mixture is 30%, between 20% and 50%.

Worked example

Example 3: An acid solution

A lab technician needs 30 L of a 30% acid solution. She has a 20% solution and a 50% solution. How much of each should she mix?

Let x be the litres of 20% solution and y the litres of 50% solution.
Total volume: x + y = 30.
Amount of acid: 0.20x + 0.50y = 0.30(30) = 9.
Solve by substitution: y = 30 − x. Then 0.20x + 0.50(30 − x) = 9, so 0.20x + 15 − 0.50x = 9, −0.30x = −6, x = 20. Then y = 10.
Check: 20 + 10 = 30 L ✓. Acid: 0.20(20) + 0.50(10) = 4 + 5 = 9 ✓, and 9 is 30% of 30 ✓.

Mix 20 L of the 20% solution with 10 L of the 50% solution.

Organize with a table

For mixture and investment problems, make a table with columns Amount, Rate (as a decimal) and Amount × Rate. One row for each part and one for the total. Then the Amount column gives the first equation and the Amount × Rate column gives the second.

Worked example

Example 4: Investing summer earnings

Ahmed invests $5000 of his savings, part in a savings account paying 3% simple interest per year and the rest in a GIC paying 5%. After one year he earns $190 in interest. How much did he put in each?

Let s be the amount at 3% and g the amount at 5%, in dollars.
Total invested: s + g = 5000. Interest: 0.03s + 0.05g = 190.
Multiply the interest equation by 100 to clear decimals: 3s + 5g = 19 000.
Multiply the first equation by 3: 3s + 3g = 15 000. Subtract: 2g = 4000, g = 2000. Then s = 3000.
Check: 3000 + 2000 = 5000 ✓. Interest: 0.03(3000) + 0.05(2000) = 90 + 100 = 190 ✓.

Ahmed put $3000 in the savings account and $2000 in the GIC.

Just for fun

Funny-but-true: if you paddle upstream at exactly the speed of the current, your ground speed is c − w = 0. You get a full workout and stay in the same spot, like a treadmill with a nicer view.

Hands-on activity

Make a juice blend

You need: a measuring cup, a pitcher, juice (any kind), water, two glasses and a grown-up. (Wipe up spills right away.)

Make a half-strength drink: 1 cup juice + 1 cup water = 2 cups that are 50% juice. Pour it into the pitcher.
Goal: make 4 cups of a 75% juice drink by mixing some of the 50% drink with some pure (100%) juice. Predict first: how much of each?
Set up a system. Let x be the cups of 50% drink and y the cups of pure juice: x + y = 4 and 0.5x + 1.0y = 0.75(4) = 3.
Solve: subtract the second equation from the first to get 0.5x = 1, so x = 2 and y = 2.
Mix 2 cups of the half-strength drink with 2 cups of pure juice. Taste a little of each drink to compare.
Challenge: how would you make 4 cups of a 60% drink from the same two drinks?

What to look for: step 6: x + y = 4 and 0.5x + y = 2.4, so 0.5x = 1.6, x = 3.2 cups of 50% drink and y = 0.8 cups of juice. The percent of the blend is always between the two starting percents.

Your turn

A motorboat travels 60 km downstream in 2 h and returns upstream in 3 h. Find the boat's speed in still water and the current's speed.

How many litres of a 10% acid solution and a 40% acid solution must be mixed to make 12 L of a 20% solution?

$8000 is invested, part at 4% and the rest at 6% simple interest. The total interest after one year is $410. How much is invested at each rate?

A store mixes almonds at $24/kg with raisins at $8/kg to make 5 kg of trail mix worth $16/kg. How much of each?

A plane flies 2400 km with the wind in 4 h and back against the wind in 5 h. Find the airspeed and wind speed.

Explain why mixing a 20% solution with a 50% solution cannot produce a 60% solution.

More practice: warm-up

A canoe paddles at 7 km/h in still water. The current is 3 km/h. What is its speed downstream? Upstream?

How much pure acid is in 15 L of a 40% acid solution?

Write 6.5% as a decimal.

What is the simple interest on $1200 at 5% for one year?

A car travels 240 km in 3 h. What is its average speed?

More practice: on your own

A kayaker paddles 15 km upstream in 3 h and 15 km back downstream in 1.5 h. Find the paddling speed and the current.

A pharmacist mixes a 2% cream with a 10% cream to make 200 g of a 5% cream. How much of each?

Ms. Okafor invested $12 000, part at 2.5% and the rest at 4%. The interest after one year was $390. How much was invested at each rate?

A coffee shop blends a $30/kg coffee with an $18/kg coffee to make 12 kg of a blend worth $22/kg. How much of each?

Two cyclists start 90 km apart and ride toward each other. One rides 6 km/h faster than the other. They meet after 2.5 h. Find their speeds.

Challenge

A radiator holds 16 L of coolant that is 30% antifreeze. How much should be drained and replaced with pure antifreeze to get a 50% mixture?

A plane takes 5 h to fly 3000 km into a headwind and 4 h to fly back with a tailwind. On another day the wind is twice as strong, in the same direction. How long would the 3000 km trip into the headwind take?

Lena invested some money at 3% and twice as much at 4.5%, simple interest. Her total interest for one year was $360. How much did she invest in total?

Check your understanding

Quiz

A boat's speed in still water is b and the current is w. Its speed going upstream is:

A canoe goes 18 km downstream in 2 h. What equation does this give?

You mix 10 L of a 30% solution with 10 L of a 50% solution. What percent is the mixture?

$6000 is invested, x at 2% and y at 5%, and earns $210 in a year. Which system fits?

Which mixture percent is impossible to make from a 15% and a 35% solution?

A plane flies at 500 km/h in still air with a 40 km/h headwind. How long does a 920 km trip take?

For parents and teachers

Big idea: in rate, mixture and investment problems, one equation adds up totals and the other adds up the ingredient (or interest, or speed). This lesson continues expectation 1.2 (solving realistic problems with linear systems).

How to help: Encourage a table (amount, rate, amount × rate) before writing equations. The juice activity makes mixture problems concrete and tasty. Talk about simple interest with real numbers from a bank's savings rate (the rates in this lesson are examples).

Common difficulties:

Adding the percents instead of the amounts of ingredient.
Forgetting to change percents to decimals (5% = 0.05).
In current/wind problems, using distance instead of speed in the equations: first find speed = distance ÷ time.
Answers outside the possible range (a mixture percent must be between the two ingredients' percents).