Reasoning with Equations & Inequalities — Print
Before you solve anything, translate. Most of the difficulty in a word problem is turning English into algebra.
What the practice questions cover
These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
"Five less than a number is twelve." Write that as an equation.
Reading it left to right gives 5 - x = 12. That is the natural thing to do, and it is wrong.
Say it aloud instead. Five less than a number — you start with the number, and five is taken off it.
So it is x - 5 = 12, and x = 17. Check: five less than 17 is indeed 12.
The wrong version gives x = -7, which also solves its own equation perfectly. Nothing in the algebra warns you.
This is why translating is the hard part, not solving. English word order and mathematical order are not the same, and "less than", "subtract from" and "fewer than" all reverse it.
Translate the words
Build the habit of writing what each letter means before anything else. "Let b = the number of blue marbles." One sentence, and half the confusion disappears.
Sum, more than, increased by → +
Difference, less than, decreased by → -
Of, product, times, each → ×
Per, quotient, shared equally → ÷
Is, was, gives, results in → =
"Less than" flips the order
"5 less than x" is x - 5, not 5 - x. Read it as "take 5 away from x". The same flip happens with "subtracted from".
Naming the unknown well
When two quantities are compared, name the smaller or simpler one x and build the other from it.
"Maria has 7 more than twice as many as Jon." Let j = Jon's amount; then Maria has 2j + 7. Choosing Jon keeps the algebra clean.
If instead you set m = Maria's, you would be forced into m - 72 for Jon — correct but messier.
The balance-scale picture
An equation is a balanced scale. Whatever you do to one side, do to the other, or it tips.
That is the entire justification for "subtract 3 from both sides". You are keeping the scale level while you clear away what you do not want.
What inequalities mean
x > 5 does not name one number — it names all of them past 5.
≥ and ≤ include the endpoint; > and < do not. On a number line, a closed dot includes, an open dot excludes.
Phrases to recognise: "at least" means ≥, "at most" means ≤, "more than" means >, "no more than" means ≤.
Does your answer make sense?
Always return to the story. If you find x = -3 chairs, or 2.4 people, something went wrong — or the answer needs rounding in the direction the story requires.
If 4.2 buses are needed, you need 5 buses. Rounding follows the situation, not the usual rule.
Worked examples
Translate and solve
Three less than four times a number is 29. Find the number.
- Let n = the number. Name it first, every time.
- "Four times a number" is 4n. Build the expression piece by piece.
- "Three less than" that is 4n - 3. Take 3 away from 4n — the order flips.
- 4n - 3 = 29 ⇒ 4n = 32 ⇒ n = 8 Add 3 to both sides, then divide by 4.
n = 8
Two related unknowns
A rope 84 cm long is cut into two pieces, one three times as long as the other. Find both lengths.
- Let s = the shorter piece. Name the simpler quantity.
- Longer piece = 3s. Build the second from the first.
- s + 3s = 84 ⇒ 4s = 84 The pieces together make the whole rope.
- s = 21, so 3s = 63 Check: 21 + 63 = 84 ✓
21 cm and 63 cm.
An inequality from a story
Devon has $50 and buys a $12 ticket. Snacks cost $4 each. How many can he buy?
- Let s = number of snacks. Name the unknown.
- 12 + 4s ≤ 50 Total spending can be at most $50.
- 4s ≤ 38 ⇒ s ≤ 9.5 Subtract 12, then divide by 4.
- He cannot buy half a snack. Round down — the story demands it.
At most 9 snacks.
Words to know
- Variable
- A letter standing for an unknown number.
- Equation
- A statement that two expressions are equal.
- Inequality
- A statement that one expression is less than or greater than another.
- Solution
- A value that makes the statement true.
Mistakes to watch for
Writing "5 less than x" as 5 - x. It is x - 5: the phrase says five is taken off x, so x must come first. "Subtract from" reverses in the same way.
Skipping the "let x = …" step. Without it you can solve correctly and still not know what the answer means — was x the number of adults, or the money they spent? Write it down before the first line.
Rounding by the usual rule instead of by what the situation allows. If 9.4 buses are needed, the answer is 10 — you cannot send four tenths of a bus. Ask what the context permits.
Answering with x when the question wanted something else. Many problems define x as one part and then ask for the total. Re-read the question after solving, every time.
Before symbols, all algebra looked like this. Problems were written as paragraphs and solved in prose — the style is called rhetorical algebra.
A 9th-century problem might read: "a square and ten roots are equal to thirty-nine dirhams." That is x2 + 10x = 39.
Every equation you write is a translation out of a language mathematics spent a thousand years escaping.
Problem 1
Which expression means "5 more than n"?
- n + 5
- 5 - n
- n - 5
- 5n
Worked answer
n + 5
n + 5
Problem 2
Which expression means "7 less than x"?
- 7 - x
- x - 7
- x + 7
- x7
Worked answer
x - 7
x - 7
A very common slip is writing 7 - x — but you are starting from x.
Problem 3
Twice a number is 18. Find the number.
Worked answer
9
2n = 18
n = 9
Problem 4
Which symbol means "at least"?
- <
- >
- ≤
- ≥
Worked answer
≥
≥
Problem 5
Three times a number, plus 2, is 17. Find the number.
Worked answer
5
3n + 2 = 17
3n = 15
n = 5
Problem 6
A number decreased by 9 is 23. Find the number.
Worked answer
32
n - 9 = 23
n = 32
Problem 7
An 84 cm rope is cut into two pieces, one three times as long as the other. Find the shorter piece.
Worked answer
21
s + 3s = 84
4s = 84
s = 21, and the longer piece is 63 ✓
Problem 8
The sum of two consecutive integers is 47. Find the smaller one.
Worked answer
23
n + (n + 1) = 47
2n + 1 = 47
2n = 46, so n = 23 and the other is 24.
Problem 9
"5 less than twice a number is 19." Find the number.
Worked answer
12
2n - 5 = 19
2n = 24
n = 12
Problem 10
Devon has $50. A ticket costs $12 and snacks are $4 each. At most how many snacks can he buy?
Worked answer
9
12 + 4s ≤ 50
4s ≤ 38
s ≤ 9.5
He cannot buy half a snack, so 9.
Problem 11
A rectangle's length is 3 more than its width, and its perimeter is 26. Find the width.
Worked answer
5
2w + 2(w + 3) = 26
4w + 6 = 26
4w = 20, so w = 5 and the length is 8 ✓
Problem 12
The sum of three consecutive even integers is 84. Find the smallest.
Worked answer
26
n + (n+2) + (n+4) = 84
3n + 6 = 84
3n = 78, so n = 26. The three are 26, 28, 30 ✓
Problem 13
Ana is 4 years older than Ben. In 3 years, their ages will sum to 30. How old is Ben now?
Worked answer
10
In 3 years: Ben is b + 3, Ana is b + 7
(b + 3) + (b + 7) = 30
2b + 10 = 30
b = 10 — Ben is 10, Ana is 14 ✓
Problem 14
A jar holds 20 coins, only nickels and dimes, worth $1.55. How many dimes are there?
Worked answer
11
5n + 10(20 - n) = 155
5n + 200 - 10n = 155
-5n = -45, so n = 9 nickels
Dimes: 20 - 9 = 11. Check: 45 + 110 = 155 ✓
Problem 15
A number is tripled, then 7 is subtracted. The result equals the number increased by 13. Find the number.
Worked answer
10
Left: 3n - 7. Right: n + 13
3n - 7 = n + 13
2n = 20
n = 10. Check: 23 = 23 ✓
Problem 16
A school has 178 students going on a trip. Each bus holds 40. How many buses are needed?
Worked answer
5
178 ÷ 40 = 4.45
Four buses hold only 160 students
Round UP: 5 buses, because the extra 18 students still need a ride.