Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
By the end of this chapter, you will be able to:
• convert between decimals, fractions, and percentages,
• find a percent of a quantity without guessing,
• separate percent of from percent change,
• compute percent increase and percent decrease correctly,
• read everyday percent language without letting it turn into fog.
Prerequisites (if this is hard, go back to Unit 0)
You need fractions and decimal place value from Unit 0. Percent means out of 100.
Why this matters
Key idea
Percent language is everywhere: discounts, taxes, inflation, test scores, concentration, growth, and error rates.
If percentages feel vague, real-world information feels vague.
Real life: what it’s for
A sale says “25% off.” A receipt says “8% sales tax.” A news report says “rent increased by 12%.” A teacher says “you scored 84%.”
Those are not four unrelated topics. They are one idea wearing four different outfits.
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Chapter 4. Decimals, percentages, and translating between them Decimal, fraction, and percent are the same number
Key idea
Unit 0 gave the base idea: percent means out of 100. This chapter builds fluency on top of that.
For example,
18
0 . 18 =
= 18%
100
These are not three different values. They are three ways to write the same value.
Recipe (steps that always work)
To move between the three forms:
1. decimal to percent: multiply by 100,
2. percent to decimal: divide by 100,
3. percent to fraction: write the percent over 100, then simplify, 4. fraction to decimal: divide numerator by denominator.
“Move the decimal point” is only a shortcut name. The real move underneath is multiply or divide by 100.
Worked example (no skipped steps)
Convert 0 . 45 to a percent.
Step 1: remember what 0 . 45 means. It means forty-five hundredths: 45
0 . 45 = 100
Step 2: write that same value as a percent. Since percent means “out of 100,”
45 = 45%
100
Shortcut view: multiplying 0 . 45 by 100 gives 45, then we attach the percent sign.
Answer: 0 . 45 = 45%.
Worked example (no skipped steps)
Convert 18% to a decimal and a fraction.
Step 1: use the meaning of percent.
18
18% = 100
Step 2: write the decimal form. Dividing by 100 moves the decimal point two places to the
45
left:
18 = 0 . 18
100
Step 3: simplify the fraction form. Both 18 and 100 are divisible by 2: 18
18 ÷ 2
9
=
=
100
100 ÷ 2
50
Answer: 18% = 0 . 18 = 9 .
50
Worked example (no skipped steps)
Convert 3 to a decimal and a percent.
20
Step 1: convert the fraction to a decimal by dividing.
3 = 3 ÷ 20 = 0 . 15
20
Step 2: convert the decimal to a percent. Multiply by 100: 0 . 15 · 100 = 15
So:
0 . 15 = 15%
Answer: 3 = 0 . 15 = 15%.
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Note
A percent can be bigger than 100%. For example,
250% = 2 . 5
That means two and a half times the original amount. So do not build the fake rule that percentages must stay between 0 and 100.
Percent of a quantity
Key idea
The phrase “𝑝% of a quantity” means:
percent as a decimal or fraction × quantity
The word “of” is the signal to multiply.
Chapter 4. Decimals, percentages, and translating between them Recipe (steps that always work)
To find 𝑝% of a quantity:
1. convert the percent to a decimal or fraction,
2. multiply by the quantity,
3. report the answer in the same units as the quantity.
Worked example (no skipped steps)
Find 12% of 90.
Step 1: convert the percent.
12% = 0 . 12
Step 2: multiply because “of” means multiply.
0 . 12 · 90 = 10 . 8
Answer: 12% of 90 is 10 . 8.
Worked example (no skipped steps)
Find 25% of 64.
Step 1: choose the cleaner form. Since
25
1
25% =
=
,
100
4
it is easier to write:
1
25% of 64 =
· 64
4
Step 2: compute the multiplication. Taking one fourth of 64 means dividing 64 by 4: 1 · 64 = 16
4
Answer: 16.
Worked example (no skipped steps)
Find 250% of 40.
Step 1: convert the percent.
250% = 2 . 5
Step 2: multiply.
2 . 5 · 40 = 100
Answer: 100.
Percent increase and percent decrease
47
Note
Do not confuse these two:
• 20% of 50 means 0 . 20 · 50 = 10,
• 20 less than 50 means 50 − 20 = 30.
A percent is a rate. It is not the same thing as a raw number.
Percent increase and percent decrease
Key idea
These are different questions:
• Percent of a quantity: how much is 𝑝% of something?
• Percent change: how much did a value grow or shrink relative to the original value?
For percent change, the original value matters. That is the whole point.
Recipe (steps that always work)
To find the new value after an increase or decrease: 1. identify the original value,
2. identify the rate as a decimal,
3. for an increase, multiply by 1 + 𝑟,
4. for a decrease, multiply by 1 − 𝑟.
To find the percent change itself :
change
percent change = original value
Then convert that decimal to a percent.
Worked example (no skipped steps)
A backpack costs $80 and is discounted by 15%. Find the sale price.
Step 1: a discount means a decrease. So we keep:
100% − 15% = 85%
As a decimal, that is
85% = 0 . 85
Step 2: multiply the original price by the remaining decimal.
0 . 85 · 80 = 68
Chapter 4. Decimals, percentages, and translating between them Answer: the sale price is $68.
Worked example (no skipped steps)
A notebook costs $25 and sales tax is 6%. Find the total cost.
Step 1: tax is an increase, so we keep the original 100% and add 6%.
100% + 6% = 106% = 1 . 06
Step 2: multiply by the original price.
1 . 06 · 25 = 26 . 50
Answer: the total cost is $26.50.
Worked example (no skipped steps)
A test score rises from 70 to 84. Find the percent increase.
Step 1: find the raw change.
84 − 70 = 14
Step 2: compare that change to the original value, which is 70.
14 = 0 . 2
70
Step 3: convert the decimal to a percent.
0 . 2 = 20%
Answer: the score increased by 20%.
Worked example (no skipped steps)
A population falls from 500 to 440. Find the percent decrease.
Step 1: find the raw change.
500 − 440 = 60
Step 2: compare that change to the original value 500.
60 = 0 . 12
500
Step 3: convert to a percent.
0 . 12 = 12%
Answer: the population decreased by 12%.
Percent increase and percent decrease
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Worked example (no skipped steps)
A phone battery is at 40% and later reaches 50%. Is that a 10% increase?
No. The battery level changed by 10 percentage points, but percent increase compares that change to the original level.
Step 1: find the raw change.
50 − 40 = 10
Step 2: divide by the original level.
10 = 0 . 25
40
Step 3: convert to a percent.
0 . 25 = 25%
Answer: it is a 25% increase, not a 10% increase.
Common mistakes (and how to avoid them)
• confusing 20% with 20,
• moving the decimal point the wrong way,
• forgetting that “of” means multiply,
• treating percent change as raw subtraction only,
• dividing by the new value instead of the original value.
Bridge (why this matters later)
Percent language returns later in two major ways.
• In statistics, percentages connect to relative frequency, error rates, and interpretation of data.
• In functions and calculus, percent change becomes an early doorway into rates of change.
If this chapter feels stable, many later interpretations become much easier.
Recap: what you’re taking away
Three questions that look similar are actually different:
• Convert it. Example: 0 . 32 to a percent.
• Find a percent of a quantity. Example: 12% of 90.
• Find percent change. Example: from 70 to 84.
If you can tell which question you are answering, the arithmetic becomes much calmer.
Chapter 4. Decimals, percentages, and translating between them Problems
4.1. Convert 0 . 08 to a percent.
4.2. Convert 62% to a decimal.
4.3. Convert 3 to a decimal and a percent.
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4.4. Convert 125% to a decimal.
4.5. Convert 0 . 375 to a fraction and a percent.
4.6. Find 15% of 80.
4.7. Find 8% of 250.
4.8. Find 25% of 96.
4.9. Find 150% of 40.
4.10. Find 12 . 5% of 64.
4.11. A $40 shirt is discounted by 30%. Find the sale price.
4.12. A $120 item has 8% sales tax. Find the total cost.
4.13. A value of 50 increases by 10%. Find the new value.
4.14. A value of 200 decreases by 8%. Find the new value.
4.15. A test score rises from 60 to 72. Find the percent increase.
4.16. A price rises from $50 to $65. Find the percent increase.
4.17. A score drops from 90 to 72. Find the percent decrease.
4.18. A population falls from 800 to 680. Find the percent decrease.
4.19. A recipe uses 20% of a 500-gram bag of sugar. How many grams is that?
4.20. A bank balance of $400 grows by 5%. What is the new balance?
4.21. A phone plan costs $50 and gets a 12% discount. What is the discounted price?
4.22. Convert 7 to a decimal and a percent.
8
4.23. Convert 2 . 4 to a percent.
4.24. A battery level goes from 25% to 40%. Find the percent increase.
Quick checklist (if you miss it, reread and try again)
• I can move between fraction, decimal, and percent forms.
• I know that “of” means multiply.
• I can tell the difference between percent of a quantity and percent change.
• I know to divide by the original value when finding percent increase or decrease.
Expressions and the distributive
property