Mathematical Operations
Mathematical Operations is a scoring topic of the reasoning paper of ADRE Grade IV. In these questions the usual symbols of arithmetic, such as plus, minus, multiply and divide, are either replaced by other symbols or given a new meaning, and the candidate has to work out the value of an expression by using the new meaning. Sometimes a wrong equation is given and the candidate must choose the correct symbols to make it true. The topic needs no difficult calculation; it only needs careful reading of the changed symbols and step by step working.
What the Topic Means
In an ordinary sum, the symbols plus, minus, multiply and divide have their usual meanings. In mathematical operations questions these meanings are deliberately changed. For example, a question may say that plus means multiply, minus means divide, multiply means plus and divide means minus. The candidate must first replace every symbol of the given expression with its new meaning and only then calculate the answer. The whole difficulty of the topic lies in not forgetting the new meaning halfway through the sum.
Types of Questions
There are three common types. In symbol substitution, the given symbols stand for other operations, and the value of an expression has to be found. In symbol correction, a statement with wrong symbols is given and the candidate must replace two symbols to make it correct. In coded operation, letters or signs stand for the four operations and an equation is to be completed. A fourth type asks the candidate to find which two numbers or symbols should be interchanged to make an equation true.
Order of Operations
The order in which operations are performed is very important. In ordinary arithmetic, division and multiplication are done before addition and subtraction, and brackets are done first of all. This order is remembered by the word BODMAS, which stands for Brackets, Of, Division, Multiplication, Addition and Subtraction. In mathematical operations questions the same order is followed, but with the new meanings of the symbols. A candidate who ignores the order of operations very often reaches a wrong answer even after using the right symbols.
Method of Solving
- Read the changed meaning of every symbol carefully.
- Rewrite the given expression with the real symbols, one below the original.
- Apply BODMAS and solve the brackets first.
- Do division and multiplication before addition and subtraction.
- Check the final answer once more against the changed meanings.
Worked Example
Suppose it is given that plus means multiply, minus means divide, multiply means plus and divide means minus. Now consider the expression 8 + 4 − 2 × 6 ÷ 3. Replacing the symbols, this becomes 8 × 4 ÷ 2 + 6 − 3. By BODMAS, first divide: 4 ÷ 2 = 2, so we get 8 × 2 + 6 − 3. Then multiply: 8 × 2 = 16, giving 16 + 6 − 3. Finally, 16 + 6 = 22 and 22 − 3 = 19. So the value of the expression is 19. This shows how each symbol must be replaced before any calculation is done.
How This Topic Appears in the Examination
Mathematical operations usually appears as two to four objective questions of one mark each. The questions are simple and can be solved in less than a minute with neat working. Some questions use symbols such as an asterisk or a triangle to stand for an operation, and the candidate has to find the value of a new expression. Others ask which two symbols must be interchanged to make an equation true. Because the arithmetic is easy and the marks are certain, this topic should never be left unattempted.
Memory Tips
Write the changed meanings in a small box at the top of your rough sheet so that you never forget them. Always rewrite the whole expression with the real symbols before calculating; this single habit prevents most mistakes. Remember BODMAS and apply it strictly. For symbol correction questions, try the options one by one instead of guessing. Keep your working neat and vertical so that the steps can be checked easily.
Common Mistakes to Avoid
- Using the old meaning of a symbol out of habit.
- Ignoring the order of operations and solving from left to right only.
- Replacing some symbols but not all of them.
- Making a small arithmetic error while dividing or multiplying.
- Forgetting that brackets must always be solved first.
How to Prepare This Topic
- Revise the four basic operations and the BODMAS rule.
- Practise writing expressions with changed symbols every day.
- Solve at least ten questions of each type daily.
- Check your answer by working backwards whenever possible.
- Time yourself so that each question takes less than a minute.
More Solved Examples
Suppose the question says, "If + means ÷, ÷ means −, − means × and × means +, find the value of 12 + 6 ÷ 2 − 3 × 4." After substitution the expression becomes 12 ÷ 6 − 2 × 3 + 4. By BODMAS we first find 12 ÷ 6 = 2 and 2 × 3 = 6, so the expression is 2 − 6 + 4 = 0. In another type a sign is missing and the candidate must insert it to make an equation true, as in "8 _ 4 _ 2 = 4", where 8 ÷ 4 + 2 works because 2 + 2 = 4, so the correct signs are ÷ and +. Always simplify the expression step by step and never skip the order of operations, because one wrong order changes the whole answer.
Advanced Tips for the Exam
Copy the expression on rough paper and replace each symbol one by one before simplifying, so that you do not lose track of the changes. Follow BODMAS strictly: brackets first, then of, then division, then multiplication, then addition and subtraction. In substitution questions, first translate the whole expression and only then calculate the value. When inserting a sign, check the answer by trying the two or three likely combinations, because the options usually contain only one correct set. Time saved on these short questions can be used for the longer arithmetic sections of the paper.
Quick Revision Checklist
- BODMAS order learnt by heart.
- Substitution of symbols practised with examples.
- Left-to-right reading of operations of equal priority.
- Inserting a missing sign to balance an equation.
- Careful copying of the expression on rough paper.
- Checking the answer by putting the values back.
Frequently Asked Questions
What does BODMAS stand for?
BODMAS stands for Brackets, Of, Division, Multiplication, Addition and Subtraction. It gives the order in which the operations of a sum must be performed.
Why must I replace every symbol before calculating?
Because if even one symbol keeps its old meaning, the whole value changes. Rewriting the expression with the real symbols makes the new meaning clear before you start.
What is a symbol correction question?
In this type a wrong equation is given and you must interchange two symbols so that the equation becomes correct. The options are tested one by one.
Are the calculations difficult in this topic?
No. The arithmetic is simple. The difficulty lies only in reading the changed symbols correctly and applying the order of operations, so the topic is easy to score.
What is the best way to avoid mistakes?
Write the changed meanings at the top of your rough sheet, rewrite the expression with the real symbols, and then solve strictly by BODMAS with neat vertical working.