B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Choosing the Best Form: Problems with Fractions, Decimals and Percents

B1.4 A1About 45 minutes

Learning goals

I can compare amounts given as fractions, decimals and percents.
I can choose the most convenient form to solve a problem.
I can solve multi-step problems that mix fractions, decimals and percents.
I can explain why two percents in a row don't simply add.

Words to know

discount
An amount taken off a price, often given as a percent or fraction: 25% off, 1/3 off.

multi-step problem
A problem that needs more than one calculation, where the answer to one step is used in the next.

a percent of a percent
Finding a percent of an amount that was already found using a percent, like 40% of 3/8 of the students.

Which deal is best?

Three stores sell the same $60 boots:

Store A: 1/3 off
Store B: 30% off
Store C: pay 0.65 of the price

The offers use three different forms. To compare, you can change them all to the same form, or just work out the price at each store. Let's find out in Example 1.

Choosing a form

There is no single best form. Choose the one that makes your problem easiest:

Fractions are great for mental math with "friendly" amounts: 1/4 of 48, 2/3 of 90. They are also exact for thirds and sixths, which have repeating decimals.
Decimals are easiest with a calculator and with money: 0.65 × 60.
Percents are easiest for comparing, because everything is out of 100: 85% vs 86%.

When a problem has several steps, the answer to one step becomes the whole for the next step.

Situation	Easiest form	Why
1/3 off a $60 price	fraction	60 ÷ 3 = 20 in your head
Price per kg on a calculator	decimal	Money is written in decimals
Comparing test scores 17/20 and 84%	percent	Both out of 100: 85% vs 84%
Making 150% of a recipe	fraction or decimal	150% = 1 1/2 = 1.5
Tiny amounts, like 0.5% interest	decimal	0.005 × amount

Pick the form that fits the job.

Worked example

Example 1: Best deal

Which store has the best price for the $60 boots?

Store A: 1/3 off. 1/3 of 60 = 20, so the price is 60 − 20 = $40.
Store B: 30% off. 30% of 60 = 18, so the price is 60 − 18 = $42.
Store C: pay 0.65 of the price. 0.65 × 60 = $39.
Compare: $39 < $40 < $42.

Store C has the best price, $39.

Each bar is the full $60 price, drawn to scale. The solid part is what you pay: Store C has the shortest solid part, $39.

Worked example

Example 2: Compare scores

Ella scored 17/20 on a quiz, Ben scored 0.86 of the marks, and Zoe scored 84%. Who did best?

Change all to percents. 17/20 = 85/100 = 85%.
0.86 = 86%.
Zoe: 84%.

Ben did best with 86%.

Worked example

Example 3: Multi-step

A school has 240 students. 3/8 of them take the bus. Of the bus riders, 40% live more than 5 km away. How many students is that? What percent of the whole school is it?

Step 1: 3/8 of 240 = 240 ÷ 8 × 3 = 30 × 3 = 90 bus riders. (A fraction is easiest here.)
Step 2: 40% of 90 = 0.4 × 90 = 36 students. (The whole is now 90, not 240.)
Step 3: 36 ÷ 240 = 0.15 = 15% of the school.
Check another way: 3/8 × 2/5 = 6/40 = 3/20 = 15%. ✓ (40% = 2/5.)

36 students, which is 15% of the school.

Worked example

Example 4: A recipe

A bannock recipe uses 2/3 cup of milk. You want to make 150% of the recipe. How much milk do you need?

150% = 3/2.
3/2 × 2/3 = 6/6 = 1.

1 cup of milk

Watch out!

Two percents in a row don't simply add. 40% off, and then 25% off the sale price, is not 65% off. The second discount is taken from a smaller amount. You pay 0.6 × 0.75 = 0.45 of the price, so it is 55% off in total.

After 40% off you pay 60% of the price. Taking 25% off that removes only 15% more of the original, so the total discount is 55%, not 65%.

Just for fun

A sign says: "50% off, then another 50% off!" Some shoppers run in expecting free stuff. But 0.5 × 0.5 = 0.25, so you still pay 25% of the price. The sign was only 75% as exciting as it looked.

Hands-on activity

Flyer showdown

You need: two or three store flyers (paper or online), a calculator, paper.

Find the same kind of item at two different stores (for example, cereal, juice or batteries), or one item with two different deals.
Write each deal in all three forms (for example, 25% off = 1/4 off = pay 0.75 of the price).
Find the actual price you would pay at each store. Include 13% HST if the item is taxed.
Look for a "buy 2, get 1 free" or "buy one, get one 50% off" deal. What percent do you save overall?

Model answers: "Buy 2, get 1 free" means you pay for 2 out of 3, which is 2/3 of the price, so you save 1/3 = 33 1/3%. "Buy one, get one 50% off" means you pay 1.5 out of 2 prices, which is 75%, so you save 25%.

Math talk: "Which form did you find easiest to compare? Did everyone in your family agree?"

Your turn

Which is greater: 3/5 or 62%?

Order from least to greatest: 0.45, 4/9, 45.5%

An $80 jacket is on sale for 25% off at one store and 1/5 off at another. Which is cheaper?

In a class of 30, 2/5 of the students walk to school. 50% of the walkers bring a lunch. How many walkers bring a lunch? What percent of the class is that?

A recipe uses 3/4 cup of oats. How much do you need for 200% of the recipe?

Jo made 18 of 24 free throws. Sami made 72% of hers. Who had the better rate?

A store has a "buy 2, get 1 free" deal. What percent do you save on the three items?

More practice: warm-up

Which is greater: 1/2 or 0.45?

Find 1/4 of 20.

Find 10% of 70.

Find 0.5 of 18.

What is the easiest way to find 50% of 64?

More practice: on your own

Order from greatest to least: 7/8, 0.86, 88%, 0.87

A $150 bike is 20% off at Store A, plus 13% HST. At Store B the same bike costs $125, tax included. Which is cheaper?

A class of 28 has 3/7 of students with a dog. 25% of the dog owners also have a cat. How many students have both?

A recipe uses 1 1/4 cups of milk for 12 pancakes. How much milk for 30 pancakes?

In Grade 8, 0.35 of the 60 students chose hockey as their favourite sport. In Grade 7, 2/5 of the 50 students did. Which grade has more hockey fans? Which has a greater percent?

Challenge

A coat is marked 40% off. At the till you get a further 25% off the sale price. Is that the same as 65% off? What is the total discount?

In a garden, 3/8 is carrots, 0.3 is beans, 20% is peas and the rest is herbs. What percent is herbs? What fraction is that?

A class wanted to raise $1 200 for a local food bank. They raised 115% of their goal. How much did they raise, and by how much did they go over?

Check your understanding

Quiz

Which deal gives the lowest price on a $40 item?

Which number is greatest?

Find 50% of 3/4 of 80.

A "buy 2, get 1 free" deal saves you:

What is 18/25 as a percent?

A $100 item is 1/4 off, and then 13% HST is added. What is the total?

For parents and teachers

Big idea: This lesson completes B1.4: using fractions, decimals and percents interchangeably and flexibly to solve a variety of problems. Students choose the form that makes each step easiest and solve multi-step problems where the whole changes from step to step.

Common difficulties:

Adding percents from different wholes (40% off then 25% off is not 65% off).
Using the original whole in a second step. In "40% of the bus riders", the whole is the bus riders, not the whole school.
Comparing "more" with "greater percent". 21 out of 60 is more people than 20 out of 50, but a smaller percent.

How to help: Compare real deals in flyers together and ask your child to explain which form they used and why. Choosing and justifying a strategy, and checking with a second method, builds the A1 processes of reasoning, selecting tools and strategies, and communicating.