In this lesson
What a percent is
A percent is a ratio whose second term is 100. The symbol % is
shorthand for "out of 100." So 35 % is 35 / 100, which is also
7 / 20, which is also 0.35.
The previous lesson built equivalent ratios. A percent is the
equivalent ratio whose second term has been scaled to 100. To
convert any ratio to a percent, scale until the second term is 100;
to convert a percent to a ratio, simplify x : 100 to lowest terms.
The five benchmarks
The Alberta curriculum names five benchmark percentages: 1, 5, 10, 25, 50. Every other percent can be built from these as a sum.
The five benchmarks
35 % is 25 % + 10 %. 75 % is 50 % + 25 %. 12 % is 10 % + 1 % + 1 %.
A student who knows the five benchmarks can compute almost any percent
mentally: find 10 % of the number, then scale it up or down by combining
the chunks.
Percents above 100 % (like 150 %) and below 1 % (like 0.5 %)
work the same way: build them from the same chunks. 150 % is
100 % + 50 %. 0.5 % is half of 1 %. A percent isn't capped at
100; 200 % doubles a value, and a 0.05 % interest rate is a
real number too.
Convert between ratio, fraction, decimal, and percent
The same number can be written four ways.
The conversion routine is short:
- Ratio → percent: scale the second term to 100.
7 : 20becomes35 : 100 = 35 %. - Percent → fraction: write as
x / 100, then simplify.35 % = 35/100 = 7/20. - Decimal → percent: multiply by 100 (shift two places right).
0.35 = 35 %. - Percent → decimal: divide by 100 (shift two places left).
35 % = 0.35.
Every form represents the same number. Picking the form that makes the next step easiest matters as much as the conversion itself.
Discounts, taxes, tips, and other applications
Percent is mostly used as a multiplier on a base number. A 30 %
discount on a $48 jacket reads 0.30 × 48 = 14.40 off, so the
sale price is $48 − $14.40 = $33.60. Tax is the same shape.
Try it
Type a price. Pick a discount. The receipt updates live.
Pick a discount
Receipt — Alberta
Discount comes off the sticker price first; GST applies to whatever is left. A till-style calculator. The student types a sticker price and picks a discount chip; a receipt-style breakdown shows the subtotal, the discount line, the after-discount subtotal, the GST 5% line, and the total. The discount comes off the sticker first; GST applies to whatever's left.
A misconception to correct: percentages don't add when they're
applied one after the other. A 30 % discount followed by 5 % GST
is not a "25 % off" deal. The discount gives 0.70 × $48.00 = $33.60;
GST is then 0.05 × $33.60 = $1.68; the total is $35.28. By
comparison, a flat 25 % off the same jacket would be
0.75 × $48.00 = $36.00, more than paying full discount plus tax.
The order matters, and the second percent applies to the smaller
subtotal, not the original sticker.
A 15 % tip is 0.15 × bill. A 5 % GST is 0.05 × pre-tax total.
A 10 % markup is 1.10 × wholesale price. A 30 % discount on a
$48 jacket leaves 0.70 × 48 = $33.60. The pattern: a percent
is a multiplier, and the multiplier is the percent divided by 100.
Percent increase and percent decrease
When a quantity changes from one value to another, the percent change describes the change as a fraction of the original:
A $48 jacket marked down to $36 has changed by 36 − 48 = −12
dollars. The percent change is −12 / 48 = −0.25 = −25 %, a
percent decrease of 25 %. The minus sign matters; it tells you
the direction.
A $48 jacket marked up to $54 has changed by +6 dollars. The
percent change is 6 / 48 = 0.125 = 12.5 %, a percent increase
of 12.5 %.
One more example with non-friendly numbers. A $3.20-per-hour
raise on top of $15.00 is 3.20 / 15.00 ≈ 21.3 %. Same formula;
the arithmetic just doesn't land on a round percent.
Worksheet
Try these to put the idea into practice. Use benchmarks for the friendly numbers, decimal multiplication for the awkward ones.
Practice · Not graded
MA.7.NUM.5Practice the idea
01 / 07
What is 25 % of 80?
Multiple choice: what is twenty-five percent of eighty? Four cards: twenty, two, two hundred, sixteen.Reference · Benchmark decompositions
30 % = 25 + 5
75 % = 50 + 25
12 % = 10 + 1 + 1
Show common mistakes
Student says
“A 30 % discount and a 5 % GST is the same as 25 % off.”
What it reveals
Treated percentages as additive when they apply sequentially. The two percents act on different bases (the discount on the original price, the tax on the discounted subtotal), so they don't combine by adding.
Targeted response
Apply them in order. 30 % off $100 → $70. Then 5 % GST on $70 → $73.50. A flat 25 % off the original would be $75. Different bases, different answers.
Student says
“200 % of 50 = 50. A percent can't be more than 100.”
What it reveals
Capped percent at 100, treating 100 % as the maximum value rather than as the original whole.
Targeted response
100 % is the original; anything above 100 % means more than the original. 200 % of 50 is `2 × 50 = 100`, twice the original. 150 % of 40 is `1.5 × 40 = 60`.
Student says
“$60 marked down to $50 is a 20 % decrease. I divided 10 by 50.”
What it reveals
Used the new value as the base for percent change instead of the original. The denominator of percent change is always the starting value, not the ending value.
Targeted response
Percent change is (new − old) / old. Here: (50 − 60) / 60 = −10/60 ≈ −16.7 %. Roughly a 17 % decrease, not 20 %. The original price is the base.
Going further
Percent shows up across later grades. In Grade 8, interest
(simple and compound) applies the percent multiplier across
multiple periods. Percent change problems compute
(new − old) / old and report the result as a percent, used for
prices, yields, and damage assessments.
Statistics in this same Grade 7 strand uses percentages to summarize sample proportions; the Grade 9 stats outcomes use them again in confidence-interval form. The conversion routines from this lesson apply unchanged in both.