Math: Decimals, Patterns & Word Problems
Ontario Mathematics Proficiency Test — study tool.
Scroll down past the quiz for the complete study guide on this topic.
Scroll down past the quiz for the complete study guide on this topic.
1
A 25 kg bag of fertilizer costs $62.25. Each hectare needs 67 kg. What is the cost for 8 hectares (must buy full bags)?
2
Omar runs 35.3 km twice, 37.8 km twice, and 17.6 km once. What is the total distance?
3
Four relay runners: 58.409s, 59.685s, 59.438s, 58.127s. The previous record is 240.201s. How much lower is their total?
4
Nathalie has $1,370, deposits $260, then pays: food $123, textbook $55, rent $800. What is her balance?
5
Order from least to greatest: 0.054, 0.1, 1.20, 1.0816
6
42 packages of seeds cost $1.35 each. What is the total?
7
The length of a classroom is 7.5 m. What is the length in millimetres?
8
Pattern A divides by 4: 36864, 9216, 2304, 576, ... Pattern B divides by 3: 26244, 8748, 2916, 972, ... Which number appears in both?
9
A pattern starts at 1,683 and decreases by 85 each term: 1683, 1598, 1513, 1428, ... What is the 9th term?
10
What is 6,905 ÷ 34?
0
out of 10
Quiz Complete!
Study Guide: Decimals, Patterns & Word Problems
DECIMAL ADDITION AND SUBTRACTION
Rule: Line up the decimal points, then add/subtract normally.
Add trailing zeros to make all numbers the same length.
Example: 35.3 + 35.3 + 37.8 + 37.8 + 17.6
= 70.6 + 75.6 + 17.6
= 163.8
Example: 240.201 - 235.659
Line up:
240.201
- 235.659
4.542
DECIMAL MULTIPLICATION
Multiply as if there are no decimals, then count total decimal places
in both numbers and put the decimal point that many places from the right.
Example: 42 × $1.35
42 × 135 = 5,670
$1.35 has 2 decimal places → $56.70
Quick check: 40 × 1.35 = 54, plus 2 × 1.35 = 2.70, total = 56.70 ✓
ORDERING DECIMALS
Add trailing zeros to make all numbers the same length, then compare
digit by digit.
Example: 0.054, 0.1, 1.20, 1.0816 Rewrite: 0.0540, 0.1000, 1.2000, 1.0816 Order: 0.0540 < 0.1000 < 1.0816 < 1.2000 COMMON MISTAKE: Thinking 0.054 > 0.1 because 54 > 1. Add the zeros: 0.054 vs 0.100. Now it's clear 54 < 100.
METRIC UNIT CONVERSION
km → m: × 1,000
m → cm: × 100
m → mm: × 1,000
cm → mm: × 10
Going the other direction? Divide instead.
Example: 7.5 m to mm
7.5 × 1,000 = 7,500 mm
ARITHMETIC SEQUENCES (PATTERNS)
An arithmetic sequence changes by the SAME amount each term.
That amount is called the COMMON DIFFERENCE.
Finding the common difference:
Subtract any term from the next one.
Example: 1683, 1598, 1513, 1428, ...
Difference = 1598 - 1683 = -85 (decreasing by 85)
Finding a specific term:
nth term = first term + (n-1) × common difference
Example: What is the 9th term?
9th = 1683 + (9-1) × (-85) = 1683 + 8 × (-85) = 1683 - 680 = 1003
GEOMETRIC SEQUENCES (MULTIPLY/DIVIDE PATTERNS)
Each term is multiplied (or divided) by the same number.
Example: Pattern A divides by 4: 36864, 9216, 2304, 576, 144, 36, 9 Example: Pattern B divides by 3: 26244, 8748, 2916, 972, 324, 108, 36, 12 To find a number in both patterns, keep extending each until they share a value. Both contain 36.
LONG DIVISION WITH REMAINDERS
Example: 6905 ÷ 34
How many times does 34 go into 69? → 2 (34 × 2 = 68)
69 - 68 = 1, bring down 0 → 10
How many times does 34 go into 10? → 0
Bring down 5 → 105
How many times does 34 go into 105? → 3 (34 × 3 = 102)
105 - 102 = 3
Answer: 203 remainder 3 (or 203 R3)
Quick check: 34 × 203 = 6,902. And 6,905 - 6,902 = 3 ✓
MULTI-STEP WORD PROBLEMS
These combine several operations. The key is breaking them into steps.
"ROUND UP" PROBLEMS (buying full units):
When you need a certain amount and can only buy in fixed sizes,
ALWAYS round UP.
Example: Need 536 kg of fertilizer. Bags are 25 kg.
536 ÷ 25 = 21.44 bags
Must round UP to 22 bags (can't buy 0.44 of a bag)
Cost: 22 × $62.25 = $1,369.50
BALANCE / MONEY PROBLEMS:
Track deposits (additions) and expenses (subtractions).
Example: Start $1,370. Deposit $260. Pay food $123, textbook $55, rent $800.
$1,370 + $260 = $1,630
$1,630 - $123 - $55 - $800 = $652
RELAY / TOTAL TIME PROBLEMS:
Add all individual times, then compare to a target.
Example: Four runners: 58.409 + 59.685 + 59.438 + 58.127 = 235.659s
Previous record: 240.201s
Difference: 240.201 - 235.659 = 4.542s faster
TIPS FOR WORD PROBLEMS
1. Read the whole question before calculating anything. 2. Identify what they're actually asking for (total? difference? how many?) 3. Break it into steps — write each step down. 4. Check if you need to round up (buying full units) or if exact is OK. 5. Check your units — are they asking for minutes or hours? m³ or cm³? 6. Use your calculator — these questions are designed to be solved with one.
KEY THINGS TO REMEMBER
Decimals: line up the decimal points for add/subtract Ordering decimals: add trailing zeros to compare m to mm: × 1,000 Arithmetic sequence nth term: first + (n-1) × difference Division with remainder: quotient and leftover (not a decimal) Round UP when buying full units Multi-step: break into smaller steps, track signs (+ and -)
