"At most" means ≤. "At least" means ≥. "More than" and "less than" exclude the endpoint.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A budget is an inequality
Spending must be at most what you have, which is ≤ rather than =.
The answer may need rounding
You cannot buy 7.4 notebooks. A budget answer rounds down, whatever the arithmetic says.
Check the boundary
Test the endpoint itself to see whether it belongs in the answer.
Translating constraint language
"At least" is ≥, "at most" is ≤, "more than" is >, "fewer than" is <. The two with "at" include the boundary; the other two do not. Getting this wrong shifts the answer by exactly one boundary case, which is often the case that matters.
Define the variable and its units
Write "let h be the number of hours worked" before modelling. A constraint on an undefined quantity cannot be checked, and in a multi-constraint problem you will lose track of which is which.
Context restricts the solution set
If h is hours worked, h ≥ 0 always, even if the algebra permits negatives. And if the answer must be whole, the mathematical range must be narrowed. Stating these restrictions is part of a complete answer.
Report the answer in words
x ≤ 12.5 becomes "she can work at most 12 full hours" once you account for whole hours. Rounding a constraint down or up depends on which direction keeps the constraint satisfied — think about it rather than rounding by habit.
Step 2: Try It Yourself
Tap and try it out.
spent and left
24
6
the budget
?
The hatched bar is the piece the story is asking for.
Step 3: Watch an Example
One step at a time.
Watch Dev Budget for Notebooks
Dev has £30 and notebooks cost £4 each.
- Step 1
Spending must be at most 30, so 4n ≤ 30.
Step 4: Your Turn
Practice makes it stick.
The Budget
Problem 1 of 2
£40 with pens at £3 each. How many pens at most?
The Minimum
Problem 2 of 2
You need at least 50 points and have 32. How many more at minimum?
From Words to Symbols
1 of 8
£25 with items at £5. How many at most?
2 of 8
£25 with items at £6. How many at most?
3 of 8
Match each phrase to its symbol.
Tap a card on the left to start.
4 of 8
5n ≤ 45. Largest whole n?
5 of 8
3n + 2 ≤ 20. Largest whole n?
6 of 8
Does "at most 10" include 10? 1 for yes, 0 for no.
7 of 8
Does "more than 10" include 10? 1 for yes, 0 for no.
8 of 8
£100 with tickets at £12. How many at most?
Step 5: Quick Check
Show what you know.
Question 1 of 1
£60 with books at £7. How many at most?
What You Learned
- "At most" and "at least" include the endpoint; "more than" and "less than" do not.
- A budget is an inequality, not an equation.
- Round to what the situation allows, not to what the arithmetic gives.