Simplification and Fractions
Simplification is the skill of reducing a long and untidy expression to a single clean value, and in the ADRE Grade IV examination it is tested both as a question in its own right and as the last quiet step of almost every other sum. The chapter has two faces. One face is the order of operations, which tells us the exact sequence in which the brackets, the division, the multiplication, the addition and the subtraction must be handled. The other face is the arithmetic of fractions and decimals, which decides whether the working stays short or grows long and error-prone. For a Grade IV candidate the questions are usually direct and the numbers are not large, but they are unforgiving, because a single change of sign or a single wrong step in the order spoils the whole answer. The good news is that the rules are few, fixed and easy to remember, and with regular practice the whole process becomes mechanical and quick.
Basic Terms
- Simplification: reducing an expression to its simplest and shortest value.
- Expression: a group of numbers joined by the signs of the operations.
- Term: a part of an expression separated by the signs of addition or subtraction.
- BODMAS: the fixed order of operations, standing for Brackets, Of, Division, Multiplication, Addition and Subtraction.
- Bar or vinculum: the line drawn over a group of terms, which acts as the strongest bracket.
- Numerator and denominator: the upper and the lower number of a fraction.
- Proper fraction: a fraction whose numerator is smaller than its denominator.
- Improper fraction: a fraction whose numerator is equal to or larger than its denominator.
- Mixed number: a whole number joined with a proper fraction.
The word Of in BODMAS is nothing but multiplication, and it carries a higher priority than the ordinary multiplication sign. Thus one third of ninety is thirty, and this is worked out before the surrounding division and multiplication are touched.
The BODMAS Order
Every simplification follows one fixed sequence. First the bar and the brackets are removed, working from the innermost bracket outwards. Then the powers and the roots are evaluated. Then the division and the multiplication are carried out from left to right. Finally the addition and the subtraction are done, again from left to right. Division and multiplication stand on the same rung of the ladder, so they are handled strictly in the order in which they appear. The same is true of addition and subtraction. Breaking this order gives a wrong answer even when every individual calculation is correct, and this is exactly why the examiner relies on it so heavily. A useful habit is to rewrite the expression on a fresh line after every stage, so that the eye is not confused by terms that have already been used.
Working with Brackets
The bar over a group of terms is the strongest bracket and must be cleared first. After that the round brackets, the curly brackets and the square brackets are opened in turn, from the innermost to the outermost. The most frequent slip occurs when a minus sign stands just before a bracket, because then every sign inside the bracket changes when the bracket is opened. A candidate should slow down for one moment whenever a minus precedes a bracket and check each sign. Keeping the content of a bracket together until it has been fully reduced also prevents the loss of a term, which is a very common waste of marks.
Simplification of Fractions
Fractions are the heart of this chapter. To add or subtract fractions, first bring them to a common denominator, for which the lowest common multiple of the denominators is the most economical choice. Then add or subtract only the numerators and keep the denominator the same, and finally reduce the result to its lowest terms. To multiply fractions, multiply the numerators together and the denominators together; to divide by a fraction, multiply by its reciprocal, that is, by the same fraction turned upside down. A mixed number should be changed into an improper fraction before any of these operations, and the answer may be turned back into a mixed number at the end if the question expects that form.
Simplification of Decimals
Decimals are simplified by treating them as fractions with denominators that are powers of ten, or by shifting the decimal point so that the numbers become whole. When decimals are to be divided, it is safest to remove the decimal point from the divisor by multiplying both the divisor and the dividend by the same power of ten, because this leaves the value of the division unchanged. When decimals are added or subtracted, the decimal points must be written one below the other so that the tenths, hundredths and thousandths fall in the correct columns. Carelessness with the position of the decimal point is the single most common reason for a wrong answer in this part.
The Word Of and Continuous Fractions
A continuous or complex fraction is one in which a fraction itself appears in the numerator, the denominator, or both. The safe method is to simplify the numerator completely, then simplify the denominator completely, and only after that divide the first result by the second. The word Of is handled at the same time, since it simply means multiplication and takes priority. When the words of the fraction are read carefully and the expression is written down in a single line, the whole difficulty disappears, and the remaining work is ordinary arithmetic.
Cancellation and Common Factors
The greatest time-saver in simplification is cancellation. In a product of fractions, a factor that appears in any numerator may be cancelled against the same factor in any denominator, even when the two do not belong to the same fraction. Writing every numerator and denominator as a product of its factors, and then striking out whatever is common, usually shrinks a frightening expression into a small number. Cancellation also guards against careless arithmetic, because the numbers left behind are smaller and the final division is often exact.
Approximation and Checking
Some questions ask only for an approximate value, and there it is a waste of time to calculate exactly. The working numbers are rounded to convenient values and the answer is chosen from the options, but this is safe only when the options are far apart. In every question, however, a quick estimate at the end is a valuable check, because it catches a misplaced decimal point or a value that is wildly wrong before any time is lost. Accuracy in simplification is not only about doing the sum; it is also about noticing when the answer cannot possibly be right.
How This Chapter Appears in the Examination
In ADRE Grade IV and other state papers, simplification appears as a direct question containing brackets, fractions and decimals, and also as the hidden final step inside average, percentage, ratio and interest problems. The direct questions are short, so the method and not the size of the numbers is what decides the score. This chapter rewards steady practice more than cleverness, and a candidate who can simplify accurately and quickly buys extra time for the harder questions later in the paper.
Common Mistakes to Avoid
- Doing the addition or the subtraction before finishing the division and the multiplication.
- Forgetting to change the signs inside a bracket that is preceded by a minus sign.
- Ignoring the bar over a group of terms, which is the strongest bracket.
- Adding fractions without first bringing them to a common denominator.
- Losing track of the decimal point while multiplying or dividing decimals.
How to Prepare This Topic
- Learn the BODMAS order until it needs no conscious thought.
- Practise removing nested brackets, paying special attention to the minus sign.
- Drill the addition, subtraction, multiplication and division of fractions.
- Practise decimals, taking care with the position of the point.
- Finish with timed sets of ten questions every day, and review every wrong step.
Frequently Asked Questions
What is the BODMAS rule and why is it used?
BODMAS fixes the order of operations as brackets first, then of which means multiplication, then division, then multiplication, then addition, and last subtraction. It is used so that one expression has only one correct value. Division and multiplication are of equal rank and are done from left to right, as are addition and subtraction.
Why do the signs change inside a bracket that follows a minus sign?
Because subtracting a whole group is the same as subtracting each member of the group one by one. So minus, open bracket, a minus b, close bracket becomes minus a plus b. This is the most common mistake in simplification questions that contain brackets.
How do I add two fractions with different denominators?
First find the lowest common multiple of the two denominators and rewrite each fraction with that common denominator. Then add only the numerators and keep the common denominator. Finally reduce the answer to its lowest terms. For example, one half plus one third becomes three sixths plus two sixths, which is five sixths.
How do I simplify a continuous fraction?
Simplify the numerator on its own, then simplify the denominator on its own, and only then divide the first result by the second. Never try to handle the whole expression in one stroke, because that is where most errors begin. Writing it clearly in a single line makes the two separate reductions much easier.
How should I prepare simplification for ADRE Grade IV?
Make the order of operations automatic, then practise brackets, and after that drill fractions and decimals until the arithmetic is quick and sure. Solve ten questions against the clock each day, and whenever an answer is wrong, find out whether the fault was in the order of operations or in the arithmetic.