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Mathematical Operations and Symbol Notations

Reasoning · Verbal Reasoning

8 Part1 worked examples10 practice questions

In ordinary arithmetic + means add and × means multiply. But the questions of this chapter are like a game: “suppose + means ×” or “let @ mean ‘is greater than’”. You have to calculate under the new rules with the same care. It is the most “straight” chapter of the exam — mistakes come from a lapse of attention, not from difficulty.

The chapter has four kinds of questions, which are its four main parts, preceded by the foundation:

  • Foundation — the BODMAS order and the language of the comparison signs (<, >, ≤, ≥, =).
  • New meanings for signs — solving an expression after “if + means − and × means ÷ …”.
  • Interchange — “if signs or numbers are interchanged, which equation will be correct?”
  • Hidden rule — seeing a few examples, spotting the rule and applying it to a new question.
  • Conclusions — drawing sure conclusions from “greater / smaller / equal” hidden in signs such as @, #, ©.

How to use this chapter

  1. Read — each part first explains the idea and then the method.
  2. Study the solved example — in the boxes marked “Example” the full solution is open, step by step. At every step ask: why was this step taken?
  3. Try it yourself — in the “Try it yourself first” questions the solution is hidden. Solve on paper first, then choose an option. If stuck, take the hint, and open the solution only at the very end.
  4. Know the level — every question is marked Basic, Standard or Advanced. Make the basic ones solid first.
NoteEvery answer has been checked by machine: in the sign-changing questions every option was actually solved, and in the conclusion questions every possible ordering was tried. The questions you solve are remembered in this browser.
1

Foundation — BODMAS and the comparison signs

The order of calculation and the language of greater-smaller-equal

The BODMAS order, the left-to-right rule for signs of the same rank, and the language of the five comparison signs and how they combine.

BODMAS — the order of calculation

Brackets, division, multiplication, addition, subtraction

When an expression has several signs, which to do first, the left-to-right rule for equal ranks and common slips.

If someone asks “2 + 3 × 4 = ?” one person says 20 (added first), another 14 (multiplied first). Mathematics has a fixed order so that everybody gets the same answer; it is called BODMAS:

LetterMeaningOrder
BBrackets — ( )first
OOf — such as “half of 8”after brackets
DDivision — ÷third
MMultiplication — ×fourth
AAddition — +fifth
SSubtraction — −last

The ladder to remember: brackets → division-multiplication → addition-subtraction. Division and multiplication are of one rank, and addition and subtraction of one rank. When signs are of the same rank go from left to right: in 20 − 8 + 3 first 20 − 8 = 12, then 12 + 3 = 15 (not 20 − 11 = 9). In the same way 36 ÷ 6 × 2 = 6 × 2 = 12 (not 36 ÷ 12 = 3).

NoteAll the questions in this chapter are built so that they give the same answer under both ways of reading — “left to right” and “strict BODMAS (D, then M, then A, then S)” — so there is no dispute about the answer. For the exam, though, the safe rule is the left-to-right one.

The way to solve: do one operation at a time and rewrite the whole expression each time. Doing several at once leads to mistakes. See the solution below:

Example 1 · Basic
Find the value: 48 ÷ 6 × 4 − 9 × 5
Solution, step by step
  1. 1Division and multiplication are of one rank and come first, left to right: 48 ÷ 6 = 8.
  2. 2The expression now: 8 × 4 − 9 × 5. Then 8 × 4 = 32.
  3. 3Now: 32 − 9 × 5. One multiplication is left: 9 × 5 = 45.
  4. 4Now only subtraction: 32 − 45 → −13.
Answer −13
Doing one operation at a time leaves very little room for error.

In brackets open the innermost bracket first. In (8 − (3 + 2)) × 4 first 3 + 2 = 5, then 8 − 5 = 3, and at the end 3 × 4 = 12. A minus sign before a bracket subtracts the value of the whole bracket, not only its first number.

BasicTry it yourself first
What is the value of 11 − (10 + 2) ÷ 4?
BasicTry it yourself first
What is the value of (14 + 9) × 6?
BasicTry it yourself first
What is the value of (11 + 7) ÷ 3 + 4 × 2?
StandardTry it yourself first
What is the value of (11 − (6 + 2)) × 8?
StandardTry it yourself first
What is the value of (8 − (4 + 3)) × 7?
StandardTry it yourself first
What is the value of 14 − (4 + 7) ÷ 11?

The comparison signs and how they combine

<, >, ≤, ≥, = and “not less than”

The five comparison signs, phrases such as “not less / not more”, and what two signs give together — a complete table.

SignMeaningAlso said as
A > BA is greater than BA is not smaller than or equal to B
A < BA is smaller than BA is not greater than or equal to B
A ≥ BA is greater than or equal to BA is not smaller than B
A ≤ BA is smaller than or equal to BA is not greater than B
A = BA and B are equalA is neither greater nor smaller than B

Keep two things in mind: (1) the signs with “or equal” (≥ and ≤) are weak — they leave room for being equal, so “definitely greater” does not follow from them. (2) The phrases with “not” work as the opposite: “not greater” means ≤.

Two signs together

If the relation of X and Y is known and also that of Y and Z, what can be said for sure about X and Z? The table below shows all 25 combinations. (It was made by checking every possible ordering.)

X ? Y \ Y ? Z>≥<≤=
X > YX > ZX > Z——X > Z
X ≥ YX > ZX ≥ Z——X ≥ Z
X < Y——X < ZX < ZX < Z
X ≤ Y——X < ZX ≤ ZX ≤ Z
X = YX > ZX ≥ ZX < ZX ≤ ZX = Z
  • The same direction (> and >, or < and <): the conclusion comes in that direction.
  • Strict + “or equal”: the conclusion is as strong as the strictest — > with ≥ gives > (one sure “greater” is enough), but ≥ with ≥ gives only ≥.
  • Opposite directions (> and <, ≥ and ≤): nothing is sure — X and Z may have any relation. This is what the “—” cells mean.
  • Equal (=) changes no sign; it only passes it on.
TrickThink of a ladder: both signs must go “up” or both “down” for you to reach the third. Where the direction changes (up and then down) the bridge is broken.
BasicTry it yourself first
Statement: Q ≥ T ≤ W < L. Which of the following conclusions is definitely true?
BasicTry it yourself first
Statement: B ≤ M ≥ L ≤ X. Which of the following conclusions is definitely true?
StandardTry it yourself first
Statement: Q ≤ N ≤ T > J. Which of the following conclusions is definitely true?
StandardTry it yourself first
Statement: Z ≤ M ≤ W > Q. Which of the following conclusions is definitely true?
2

New meanings for the signs

Type 1 — “if + means ×, then …”

Solving an expression after giving the signs new meanings, letters standing for signs, and picking the correct equation.

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Solving after replacing the signs

First replace, then solve

How to put the new meanings in, the swap of two signs and all four signs shifted around, and the slips to avoid.

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When letters stand for signs

“P denotes ×, Q denotes +”

Questions with letters: writing the sign for every letter, building the expression and solving it.

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Picking the correct equation and filling the blanks

Which of four equations is correct; which signs to put in

Checking every option under the changed meaning, and the “try and see” method for putting signs in blanks.

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3

Interchange of signs and numbers

Type 2 — “if + and × are interchanged, which equation will be correct?”

Interchanging two signs or two numbers to pick the correct equation, and finding the interchange that makes an equation correct.

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Interchanging to pick the correct equation

Signs, numbers or both

Solving every option after the interchange; changing the number everywhere in the interchange of numbers.

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The interchange that corrects an equation

“Which two signs must be interchanged?”

The reverse question: the equation is wrong and you have to find the interchange that makes it correct — checking all six pairs.

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4

Finding the hidden rule

Type 3 — “if 5 ★ 4 = 21, 6 ★ 3 = 15 …, then 7 ★ 2 = ?”

Recognising the rule of a new sign from examples and applying it to a new question; rules that work on the digits of two-digit numbers.

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The rule of two numbers

Sum, difference, product, square — and their mixes

The order for finding the rule, the list of common rules and making sure by checking.

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Rules that work on the digits of two-digit numbers

“64 ★ 52 = …” — break it into digits

When the numbers have two digits the rule often works on the digits: their sum, product, difference or a mix of tens and units digits.

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5

Conclusions from comparison symbols

Type 4 — the “greater / smaller / equal” hidden in @, #, ©

Opening the symbols into real signs, joining the statements, the test of “definitely true” and the rule of “either I or II”.

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The test of “definitely true”

True in every possible situation

Three kinds of conclusion — surely true, surely false and “may or may not be” — and building a chain in one direction.

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Two conclusions and five options

Only I, only II, either I or II, neither, both

The meaning of the five options, the rule of “either I or II” and practice.

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Coded-symbol questions (@, #, ©)

First open the symbols into real signs

The real exam form: a table of symbols with statements and conclusions. The order of opening them and practice.

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6

Strategy guide

Spot the type, then solve; the slips to avoid

A table for spotting the four types of question and the common traps.

Spotting the type

At first sight, which type is this question

The first sign of each type and its first step.

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Common traps

The mistakes that keep happening

The commonest reasons for losing marks in this chapter and how to avoid them.

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7

Mixed practice

All types mixed — first spot the type

Questions of different types together, in three levels.

Basic practice

One rule, one step

BODMAS, the swap of two signs, simple rules and straight conclusions.

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Standard practice

Four signs shifted, interchange of numbers, rules from three examples and two conclusions

The standard level of every type.

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Advanced practice

Five numbers, double interchange, conclusions with five variables

Long expressions, both kinds of interchange, rules on digits and symbol questions with five letters.

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8

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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