Lesson Guide
Explanation
Halving ten dollars three times gives five, two fifty, one twenty-five. If the question gives you the end instead and asks for the start, reverse it: double three times. The rule is to undo the last step first and use the opposite operation each time — times becomes divide, plus becomes minus. Then run the chain forwards from your answer to check, which takes five seconds and catches nearly every slip.
Your Turn
$10 halved three times leaves how much? Check: $1.25.
Watch Out
Undo in reverse order. Reversing the first step first gives a different, wrong number.
Explanation
When two equations come together, look for the one with only a single unknown in it and solve that one first. Nine times the square is a hundred and eight, so the square is twelve. Then substitute that into the other: the triangle plus twelve is thirty-six, so the triangle is twenty-four. Check both equations at the end, and read which symbol the question actually asks for — it is often the one you did second.
Your Turn
7 × □ = 84 and △ + □ = 30. Find △. Check: □ = 12, so △ = 18.
Watch Out
Read which symbol is wanted. Solving for the square and answering with it is the usual trap.
Explanation
Sometimes neither equation has a single unknown, but one is contained inside the other. Two stars and a heart make twenty; one star and a heart make thirteen. The difference between the rows is exactly one extra star, so a star is twenty take thirteen, seven. Then seven plus a heart is thirteen, so a heart is six, and the first row checks at twenty. Compare the rows and ask what is different between them.
Your Turn
★ + ★ + ♥ = 20 and ★ + ♥ = 13. What is ♥? Check: 6.
Watch Out
Subtract the whole row from the whole row — both sides. 20 − 13 on the right, and the symbols on the left.
Explanation
A division with a remainder can always be rewritten as a multiplication plus the remainder. A hundred and seventeen shared between fourteen is eight each with five left over, which is the same as saying a hundred and seventeen equals fourteen times eight plus five. Check by working it out — fourteen eights are a hundred and twelve, plus five is a hundred and seventeen. And the remainder must always be smaller than the number you divided by, which rules out several of the offered sentences straight away.
Your Turn
Write "100 shared between 12, with 4 left over" as a number sentence. Check: 100 = 12 × 8 + 4.
Watch Out
The remainder must be less than the divisor. 117 = 14 × 7 + 19 is arithmetically true but is not a valid remainder.
Explanation
Four shapes and four techniques. A chain of operations with the end given means undoing it backwards. Two equations where one has a single unknown means solve that and substitute. Two equations that share symbols means subtracting the rows. And a list of candidate sentences means testing each one by working it out. Whichever you use, put the answer back into the original to check. Quiz and exercises now.
Your Turn
A number is doubled, then 6 is added, giving 20. What was it? Check: (20 − 6) ÷ 2 = 7.
Watch Out
With five options in front of you, testing them is often faster than solving — especially on a symbol question.
📌 Key Points — this lesson in a nutshell
- To undo a chain, start at the answer and reverse each step in the opposite order: × becomes ÷, + becomes −.
- With two symbol equations, start with the one that has only one unknown in it.
- Once a symbol's value is known, substitute it into the other equation.
- When two equations share symbols, subtracting one from the other can remove a symbol entirely.
Step 1: Introduction
Step 1 of 5
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