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:
• translate a verbal problem into algebra,
• choose a variable sensibly,
• build an equation from the situation,
• solve and interpret the answer in context,
• recognize common word-problem families instead of treating each problem like a surprise.
Prerequisites (if this is hard, go back to Unit 0)
You need equations, percentages, formulas, and checking answers against context.
Why this matters
Key idea
Word problems feel hard because they mix language and algebra.
The trick is not “being clever.” The trick is translating calmly, one piece at a time.
Real life: what it’s for
Outside a math classroom, problems do not arrive already translated into symbols. They arrive as prices, distances, scores, averages, dimensions, and constraints.
Word problems are where algebra becomes useful instead of decorative.
Recipe (steps that always work)
For most basic word problems:
1. decide what the unknown is,
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Family 1: translate-a-number sentences
109
2. choose a variable for it,
3. translate the relationships into an equation,
4. solve the equation,
5. answer in words, not only in symbols,
6. do a sanity check.
Note
A good translation is faithful, not fancy. You are not trying to impress the page. You are trying to write exactly what the story says, no more and no less.
Family 1: translate-a-number sentences
Key idea
Many word problems are really sentence translation.
Clue words help:
• sum → add,
• difference → subtract,
• twice → multiply by 2,
• is → equals.
Worked example (no skipped steps)
The sum of a number and 8 is 23. Find the number.
Let the number be 𝑥.
“The sum of a number and 8” becomes
𝑥 + 8
“is 23” means equals 23. So the equation is
𝑥 + 8 = 23
Subtract 8:
𝑥 = 15
Why this equation? The words “sum” and “is” told us to build an addition statement equal to 23.
Answer: the number is 15.
Chapter 13. Word problems without guessing
Worked example (no skipped steps)
Twice a number minus 5 is 17. Find the number.
Let the number be 𝑥.
“Twice a number” is
2 𝑥
“minus 5” gives
2 𝑥 − 5
“is 17” gives the equation
2 𝑥 − 5 = 17
Add 5:
2 𝑥 = 22
Divide by 2:
𝑥 = 11
Why this equation? The clue words were “twice,” “minus,” and “is.”
Answer: the number is 11.
Family 2: cost and rate models
Key idea
Cost problems often have this form:
total = fixed part + rate × amount
That is the same algebra pattern over and over.
Worked example (no skipped steps)
A gym charges a one-time sign-up fee of $20 and then $15 per month. If the total cost is $95, how many months is that?
Let 𝑚 be the number of months.
Fixed fee + monthly cost gives:
20 + 15 𝑚 = 95
Subtract 20:
15 𝑚 = 75
Divide by 15:
𝑚 = 5
Why this equation? The phrase “one-time fee” gave the constant term, and “per month”
gave the coefficient of the variable.
Answer: 5 months.
Family 3: geometry and perimeter
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Worked example (no skipped steps)
A taxi charges a base fee of $4 plus $3 per mile. If the total bill was $25, how many miles were traveled?
Let 𝑚 be the number of miles.
Base fee + per-mile cost gives:
4 + 3 𝑚 = 25
Subtract 4:
3 𝑚 = 21
Divide by 3:
𝑚 = 7
Answer: 7 miles.
Family 3: geometry and perimeter
Key idea
Geometry word problems often turn English labels into a formula you already know.
If the formula is clear, the story becomes much easier.
Chapter 13. Word problems without guessing
Worked example (no skipped steps)
A rectangle has length 9 and perimeter 26. What is the width?
Let the width be 𝑤.
For a rectangle,
𝑃 = 2 𝑙 + 2 𝑤
Substitute the known values:
26 = 2(9) + 2 𝑤
Use the arithmetic from Unit 0. Since
2(9) = 18 ,
the equation becomes:
26 = 18 + 2 𝑤
Subtract 18:
8 = 2 𝑤
Divide by 2:
𝑤 = 4
Why this equation? The word “perimeter” told us to start from the perimeter formula.
Answer: the width is 4.
Family 4: averages
Key idea
Average problems usually start from
sum of values
average = number of values
To solve the problem, turn that sentence into an equation.
Worked example (no skipped steps)
A student scored 68, 74, and 82 on three tests. What score is needed on the fourth test to average 75?
Let the fourth score be 𝑥.
An average of 75 over 4 tests means
68 + 74 + 82 + 𝑥 = 75
4
Add the known scores:
68 + 74 + 82 = 224
So:
224 + 𝑥 = 75
4
Family 5: comparison and consecutive numbers 113
Multiply both sides by 4 to clear the denominator:
224 + 𝑥
4 ·
= 4 · 75
4
On the left, the 4 in the numerator and the 4 in the denominator cancel: 224 + 𝑥 = 300
Subtract 224:
𝑥 = 76
Why this equation? The clue word was “average.”
Answer: the needed score is 76.
Family 5: comparison and consecutive numbers
Key idea
If one quantity is described in terms of another, define one variable and write the second quantity from it.
That is usually cleaner than inventing two unrelated variables.
Worked example (no skipped steps)
Two consecutive integers have sum 31. Find them.
Let the first integer be 𝑛. Then the next consecutive integer is: 𝑛 + 1
Their sum is 31, so:
𝑛 + ( 𝑛 + 1) = 31
Combine like terms:
2 𝑛 + 1 = 31
Subtract 1:
2 𝑛 = 30
Divide by 2:
𝑛 = 15
So the integers are:
15
and
16
Answer: the integers are 15 and 16.
Chapter 13. Word problems without guessing
Worked example (no skipped steps)
A rope 36 meters long is cut into two pieces. One piece is 8 meters longer than the other.
Find both lengths.
Let the shorter piece be 𝑥. Then the longer piece is: 𝑥 + 8
Their total is 36, so:
𝑥 + ( 𝑥 + 8) = 36
Combine like terms:
2 𝑥 + 8 = 36
Subtract 8:
2 𝑥 = 28
Divide by 2:
𝑥 = 14
So the longer piece is:
14 + 8 = 22
Answer: the pieces are 14 meters and 22 meters.
Family 6: percent stories
Key idea
Percent story problems usually ask for one of two things:
• a percent of an amount, such as a discount or tax,
• a percent change, such as an increase or decrease from an original value.
The key question is: what is the original amount?
Worked example (no skipped steps)
A shirt is discounted 20% from $45. What is the sale price?
Step 1: a 20% discount means you do not pay 20% of the price. You pay the remaining 80%.
Step 2: write 80% as a decimal.
80% = 0 . 80
Step 3: multiply by the original price.
0 . 80 · 45 = 36
Step 4: check the story. A discount should make the price lower than $45, and $36 is lower than $45.
Answer: the sale price is $36.
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Note
A good sanity check is to ask whether the answer fits the story.
If a gym plan charges $15 per month and your answer is −3 months, the algebra may be legal, but the context tells you something is wrong.
Bridge (why this matters later)
Word problems prepare you for every later book because later math is full of hidden translation.
• Functions asks you to translate descriptions into symbolic relationships.
• Calculus asks you to translate rates and changing quantities.
• Statistics asks you to translate data language into formulas and interpretations.
If you can translate calmly, later topics become much more manageable.
Common mistakes (and how to avoid them)
• choosing a variable and then forgetting what it means,
• translating too quickly and missing a sign or keyword,
• stopping at 𝑥 = 11 without answering the actual question,
• failing to check whether the result makes sense in context.
Recap: what you’re taking away
Word problems are not a separate species of math. They are algebra plus translation.
When the wording feels messy, slow down and ask:
• What is the unknown?
• What relationship does the sentence describe?
• What equation says exactly that and nothing more?
Problems
13.1. The sum of a number and 9 is 20. Find the number.
13.2. Three times a number is 27. Find the number.
13.3. Twice a number plus 4 is 18. Find the number.
13.4. A number decreased by 7 gives 12. Find the number.
13.5. A number divided by 4 equals 7. Find the number.
13.6. Five more than a number is 17. Find the number.
Chapter 13. Word problems without guessing
13.7. A taxi charges a base fee of $4 plus $3 per mile. If the total bill was $25, how many miles were traveled?
13.8. A phone plan costs $30 plus $5 per extra GB. If the total bill is $55, how many extra GB
were used?
13.9. A movie ticket costs $12 and popcorn costs $5. If one popcorn and some tickets cost $41
total, how many tickets were bought?
13.10. A gym charges $20 to join and $15 per month. If the total paid is $110, how many months is that?
13.11. A rectangle has length 8 and perimeter 26. What is the width?
13.12. A rectangle has width 5 and perimeter 34. What is the length?
13.13. The perimeter of a square is 44. What is the side length?
13.14. A student scored 68, 74, and 82 on three tests. What score is needed on the fourth test to average 75?
13.15. Two quiz scores are 70 and 82. What score is needed on the third quiz to average 80?
13.16. Three numbers have average 12. Their sum is what?
13.17. Two consecutive integers have sum 31. Find them.
13.18. Two consecutive even integers have sum 42. Find them.
13.19. A rope 36 meters long is cut into two pieces. One piece is 8 meters longer than the other.
Find both lengths.
13.20. One number is 6 more than another. Their sum is 30. Find both numbers.
13.21. A shirt is discounted 20% from $45. What is the sale price?
13.22. A price of $80 increases by 15%. What is the new price?
13.23. A car travels at 60 miles per hour for 2 . 5 hours. How far does it go?
13.24. A water tank has 12 liters at first and then gains 4 liters per minute. After how many minutes will it hold 40 liters?
Quick checklist (if you miss it, reread and try again)
• I can choose a variable and say what it means.
• I can translate common phrases into algebra.
• I can identify the family of problem I am solving.
• I can answer the question in words after solving.
Mixed review: the algebra toolkit in
one place