Concepts · Mathematical operations
When a question changes what the signs mean, translate the whole line first. Calculate after.
Riya meets this in a mock test:
If + means × and × means +, find 6 + 4 × 3.
She works on the printed line itself. The + means ×, so she changes it: 6 × 4 × 3. The × means +, so she changes the ×s: 6 + 4 + 3. That makes 13, and 13 is one of the options. She ticks it, but something about it bothers her.
The next morning the class teacher tells Riya, the class monitor, that this week the Monday and Tuesday tests have swapped. On the board, the dates read: Maths test on Monday, Hindi test on Tuesday, Art test on Monday.
Riya rubs out every Monday and writes Tuesday. Then she rubs out every Tuesday and writes Monday. She steps back. All three tests are on Monday.
Her second change undid her first: every Tuesday she had just written was turned straight back into Monday. It is what she did in the mock, where the × she had just written got changed again. So she starts over, and this time she writes the new dates underneath the old ones. She reads each old day once and writes its new day below: Maths, Monday, so Tuesday. Hindi, Tuesday, so Monday. Art, Monday, so Tuesday. She never reads what she has just written.
Write the new line underneath the printed one. Read each printed sign once and write what it means; brackets stay where they are. Calculate nothing until the new line is finished. Then solve it in the usual order: brackets, powers, then ÷ and × from left to right, then + and − from left to right.
That keeps two mistakes away: changing a sign you have already changed, and calculating with a sign you have not changed yet. The swap changes which signs are in the line. It never changes the order in which they are worked out.
The order itself: BODMAS.
13 came from changing a sign twice. 18 comes from not changing the signs at all: 6 + 4 × 3 worked as printed is 6 + 12 = 18. Either can sit among the options.
Some questions swap signs in pairs: “+ and − are interchanged, and so are × and ÷”. Others use letters: “P means +, Q means ×”. The move is the same. 12 P 3 Q 4 becomes 12 + 3 × 4 underneath; then 3 × 4 = 12, and 12 + 12 = 24.
Check which way the swap runs. “+ means ×” turns a printed + into ×. “+ is written as ×” runs the other way: a printed × stands for +.
A sign-swapping question needs two jobs done: reading the swap, and the order of operations. Writing the new line underneath finishes the first job completely before the second one starts.
1. If × means + and + means ×, find the value of 5 × 3 + 2.
Write the new line under the printed one first, reading each printed sign once.
11
New line: 5 + 3 × 2.
BODMAS: 3 × 2 = 6, then 5 + 6 = 11. (17 uses the printed signs; 30 comes from changing a sign twice; 16 goes left to right on the new line.)
2. If P means +, Q means −, R means × and S means ÷, find the value of 18 S 3 R 4 P 2.
Swap every letter for its sign before you calculate anything.
26
New line: 18 ÷ 3 × 4 + 2.
÷ and × left to right: 18 ÷ 3 = 6, then 6 × 4 = 24. Then 24 + 2 = 26. (3.5 comes from doing × before ÷.)
3. If + and − are interchanged, and × and ÷ are interchanged, find the value of 12 × 4 − 6 ÷ 2.
Every sign in the line changes. Write the whole new line before you work anything out.
15
New line: 12 ÷ 4 + 6 × 2.
÷ and × first: 12 ÷ 4 = 3 and 6 × 2 = 12. Then 3 + 12 = 15. (45 uses the printed signs: 48 − 3. 18 goes left to right on the new line.)
A week later:
If − means ÷ and ÷ means −, find 20 − 4 ÷ 3.
Riya writes the new line underneath before she touches a number: 20 ÷ 4 − 3. Then 5 − 3. She writes 2.
Every question in the app is answered before its solution shows.
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