🔢 Lesson 3.1: Numbers & Word Problems
Welcome to Mathematical Reasoning — the subject people fear most and feel proudest to conquer. This first math lesson reviews the number skills the test leans on: the order of operations, fractions, decimals, percents, and ratios, plus a reliable method for turning word problems into math. Take it slowly; every idea here builds on the last.
📚 What You'll Learn
By the end of this lesson, you will be able to:
- Use the order of operations to solve multi-step problems
- Work with fractions, decimals, and percents and convert between them
- Solve ratio and proportion problems
- Turn a word problem into an equation and solve it
⏱️ Estimated Time: 60–75 minutes (go at your own pace — there's no clock on you)
🎯 Project: Solve five real-world number problems (a tip, a discount, a unit price, a proportion, and a multi-step total), showing each step.
In This Lesson
Math You Can Rebuild
If math has been a sore spot, here's the reframe that changes everything: math is not about being "a math person." It's a set of steps you can learn, practice, and rebuild — like a recipe. Nobody is born knowing that 25% off means multiplying by 0.75; they learned it, and so can you. This module takes the everyday math from your Math & Numeracy course and sharpens it to test level, one step at a time.
Two pieces of good news about the math test: a large part of it lets you use an on-screen calculator, and many questions are really word problems — everyday situations about money, time, and measurement, which your life experience helps you understand. You've got more going for you here than it may feel like.
🧠 Mindset
Math anxiety is real and common, and it is not a sign you can't do math. Often it comes from being rushed or shamed years ago — not from any lack of ability. Give yourself permission to go slowly, to write out every step, and to make mistakes (they show you exactly what to practice). Steady, patient work rebuilds math skill at any age. You can do this, one step at a time. 🌱
Order of Operations
When a problem has several operations, you must do them in the right order, or you'll get the wrong answer. The order is often remembered as PEMDAS:
| Step | Do this |
|---|---|
| P | Parentheses ( ) first |
| E | Exponents (powers, like 4²) |
| MD | Multiply and Divide, left to right |
| AS | Add and Subtract, left to right |
🧮 Worked Example: 3 + 4 × 2² − 6 ÷ 3
- Exponent: 2² = 4, giving 3 + 4 × 4 − 6 ÷ 3
- Multiply & divide (left to right): 4 × 4 = 16 and 6 ÷ 3 = 2, giving 3 + 16 − 2
- Add & subtract (left to right): 3 + 16 = 19, then 19 − 2 = 17
Answer: 17. Do it in the wrong order (say, left to right ignoring the rules) and you'd get a different, wrong number.
💡 The calculator follows PEMDAS too
The on-screen calculator respects the order of operations, so typing 3 + 4 × 2^2 − 6 ÷ 3 gives 17 — if you enter it correctly. Understanding the order helps you type it right and catch a wrong-looking answer.
Fractions, Decimals & Percents
These three are just different outfits for the same idea — part of a whole — and the test moves freely between them. Know the key conversions:
| Fraction | Decimal | Percent |
|---|---|---|
| 12 | 0.5 | 50% |
| 14 | 0.25 | 25% |
| 34 | 0.75 | 75% |
| 110 | 0.1 | 10% |
| 15 | 0.2 | 20% |
To convert: a fraction → decimal by dividing top by bottom (1 ÷ 4 = 0.25); a decimal → percent by moving the point two places right (0.25 → 25%); a percent → decimal by moving it two places left (25% → 0.25).
🧮 Worked Example: Add 23 + 14
- Find a common denominator — the smallest number both 3 and 4 divide into is 12.
- Rewrite: 23 = 812 and 14 = 312.
- Add the tops: 8 + 3 = 11, over 12. Answer: 1112.
The most common percent task is "percent of a number" — and the trick is that "of" means multiply. Change the percent to a decimal and multiply:
🧮 Worked Example: What is 20% of 60?
- 20% as a decimal is 0.20.
- "of 60" means × 60: 0.20 × 60 = 12.
So 20% of 60 is 12. (Check: 10% of 60 is 6, and 20% is double that — 12. ✓)
Ratios & Proportions
A ratio compares two amounts — "2 cups of flour to 3 cups of water" is the ratio 2 to 3. A proportion says two ratios are equal, and it's how you scale a recipe, a map, or a price up or down. Solve proportions by cross-multiplying.
🧮 Worked Example: A recipe uses 2 cups of flour for every 3 cups of water. How much flour for 9 cups of water?
- Set up equal ratios (flour over water): 2 / 3 = x / 9.
- Cross-multiply: 2 × 9 = 3 × x, so 18 = 3x.
- Divide both sides by 3: x = 6.
You need 6 cups of flour. (Check: 9 cups of water is 3 times the recipe's 3 cups, so flour is 3 × 2 = 6. ✓)
✅ Unit rate: the everyday shortcut
A unit rate is the amount "per one" — miles per gallon, price per item. Find it by dividing, then multiply as needed. If a car goes 120 miles on 4 gallons, that's 120 ÷ 4 = 30 miles per gallon; on 10 gallons it goes 30 × 10 = 300 miles. Unit rates make comparison-shopping and proportion problems quick.
Cracking Word Problems
Most math questions are word problems. The skill is translating words into math. Certain words are signals:
| Words | Operation |
|---|---|
| total, sum, combined, altogether, more than | Add (+) |
| difference, less than, remaining, left, decreased by | Subtract (−) |
| of, product, times, each (a rate) | Multiply (×) |
| per, split, shared equally, divided, each (to find one) | Divide (÷) |
| is, equals, will be, results in | Equals (=) |
A four-step approach cracks almost any word problem:
🧮 Worked Example: A worker earns $15 per hour and works 32 hours. A $12 uniform fee is taken out. What is the take-home pay?
- Question: take-home pay.
- Numbers: $15/hour, 32 hours, $12 fee.
- Translate: "per hour" × hours, then subtract the fee: 15 × 32 − 12.
- Solve (order of operations — multiply first): 15 × 32 = 480, then 480 − 12 = $468.
Take-home pay is $468. This ties straight to the paystub math from your Math & Numeracy course.
Practice & Project
A $50 jacket is on sale for 30% off. What is the sale price?
- $15
- $20
- $35
- $47
✅ See the reasoning
The discount is 30% of $50: 0.30 × 50 = 15. Sale price = $50 − $15 = $35. (Shortcut: 30% off means you pay 70%, and 0.70 × 50 = 35.) Eliminate $47 (barely any discount) and $15/$20 (too much off) to zero in fast.
If 3 apples cost $1.50, how much do 5 apples cost (same price each)?
- $2.00
- $2.50
- $3.00
- $4.50
✅ See the reasoning
Find the unit price: 1.50 ÷ 3 = 0.50 per apple. Then 5 apples cost 0.50 × 5 = 2.50. Answer: $2.50. (By proportion: 3 / 1.50 = 5 / x → cross-multiply → 3x = 7.50 → x = 2.50. ✓)
Evaluate: 2 × (5 + 3) − 4
- 10
- 12
- 14
- 18
✅ See the reasoning
Parentheses first: 5 + 3 = 8, giving 2 × 8 − 4. Multiply: 2 × 8 = 16. Subtract: 16 − 4 = 12. Skipping the parentheses (e.g. 2 × 5 first) gives a wrong answer — order matters.
🎯 Your Project: Five Real Problems
Solve these, showing each step. Then check your work.
- Tip: Find a 15% tip on a $40 meal.
- Discount: A $60 item is 25% off — what's the sale price?
- Unit price: 12 eggs cost $3.60 — what's the price per egg?
- Proportion: A car goes 150 miles on 5 gallons — how far on 8 gallons?
- Multi-step: You buy 3 shirts at $12 each and pay with a $50 bill — how much change?
✅ Answers (check after you try)
1) 10% of 40 is 4, 5% is 2, so 15% = $6. 2) 25% of 60 = 15, sale = $45 (or 0.75 × 60 = 45). 3) 3.60 ÷ 12 = $0.30 per egg. 4) 150 ÷ 5 = 30 mpg, × 8 = 240 miles. 5) 3 × 12 = 36, then 50 − 36 = $14 change.
✅ Project Completion Checklist
- ☐ I solved all five, showing my steps
- ☐ I used "of means multiply" for the percents
- ☐ I found a unit rate for the price and the miles
- ☐ I checked my answers against the key
👥 Working with a tutor or group?
Work problems on a shared board, each person narrating their steps out loud — hearing how someone else translates a word problem often unlocks it. Then invent problems for each other from real life ("split this pretend bill," "figure this sale price"). Explaining a solution to someone else is one of the fastest ways to truly own the method.
🎯 Quick Quiz
Question 1: In a word problem, the word "of" (as in "20% of 60") usually signals which operation?
Question 2: What is 6 + 2 × 5 ?
Tips & Common Mix-Ups
✅ Do's
- Follow PEMDAS on every multi-step problem; write out each step.
- Translate keywords — "of" = multiply, "per" = divide, "is" = equals.
- Check with an estimate — does your answer make rough sense?
❌ Common Mix-Ups
⚠️ Watch Out
- Adding fractions without a common denominator: you can't add 23 + 14 as "3/7." Find the common denominator first.
- Percent decimal slips: 25% is 0.25, not 25 or 2.5. Move the point two places.
- Answering the wrong question: a discount problem may ask for the sale price, not the discount amount. Re-read what's asked.
- Calculator over-trust: a mistyped entry gives a confident wrong answer. Estimate first so a wild result stands out.
✅ Affirmation
A wrong answer in math is not a verdict — it's a clue that points to the exact step to practice. Show your work, and mistakes become your best teacher. You are rebuilding real skill.
📓 Learning Journal
Keep your study journal going. After this lesson, take five minutes to write down:
- A method you learned (PEMDAS, cross-multiplying, keyword translation)
- What clicked for you
- Your trickiest topic, to practice more
- A real place you'll use this (a tip, a sale, a recipe)
- How you feel about math today
✍️ This lesson's prompt: Write down one real number from your week — a price, a bill, a measurement — and make up a small problem with it (a tip, a discount, a split), then solve it. Turning your own life into practice makes math stick and feel useful.
📝 Lesson Summary
🎓 Key Takeaways
- Order of operations (PEMDAS): parentheses, exponents, multiply/divide, then add/subtract.
- Fractions, decimals, percents are the same idea; "percent of a number" = decimal × number.
- Proportions solve by cross-multiplying; unit rates (per one) are a fast shortcut.
- Word problems: read for the question, pull the numbers, translate keywords, solve, and check.
🎉 What You've Accomplished
You just rebuilt the number foundation the whole math test stands on — and you did it by steps, not magic. You can handle order of operations, move between fractions/decimals/percents, solve proportions, and crack word problems by translating them into math. That's a huge portion of the math subject, and you met it head-on. Be proud of it. 🎉
❓ Common Questions at This Stage
Do I really get a calculator?
For most of the math test, yes — an on-screen calculator (and a formula sheet is provided). A short first section is no-calculator, so being comfortable with basic mental math and these methods still matters. Practice with the official practice test to know the calculator's buttons.
Fractions still scare me. What should I focus on?
Master three moves: common denominators to add/subtract, multiply straight across (top×top, bottom×bottom), and fraction → decimal by dividing. On the calculator section you can often convert to decimals and work from there. Practice a few each day.
How do I get faster at word problems?
Practice the four steps until they're automatic, and build your keyword instincts ("of" = ×, "per" = ÷). Speed comes from familiarity, not rushing — and reading carefully prevents the most common error: solving the wrong question.
🔭 Looking Ahead
In Lesson 3.2, we step into algebra — but gently. You'll learn what a variable is, how to solve simple equations, read the coordinate plane, and use the formula sheet. It's less scary than it sounds, and these number skills lead right into it.
✅ Before the Next Lesson
- Do the five-problem project and check your answers.
- Practice "percent of a number" on three real prices (a tip, a tax, a sale).
- Write your Learning Journal entry.
🌟 Encouragement for the Journey
You just faced the subject people fear most — and found it's a set of clear, learnable steps, not a mystery. Every problem you work by hand rebuilds real confidence. Math isn't something you "are" or "aren't"; it's something you do, a step at a time. Keep showing your work, keep checking, and keep going. See you in Lesson 3.2! 👋
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