Simplification and Fractions for the Assam Police Constable Exam
Few candidates lose marks in the SLPRB Assam Police Constable (AB & UB) paper because they did not know a formula; far more lose marks because a long expression was mishandled. Simplification is the chapter that decides how quickly and how accurately the rest of the Elementary Mathematics paper is finished, because almost every sum, whether it is an average, a percentage, a ratio, a profit or an interest, ends in one short calculation. If that final calculation is clumsy, the earlier correct work is wasted. This chapter therefore teaches two things at once: the fixed order in which the operations must be carried out, and the lightest way to handle fractions and decimals. The method followed here is the one in the Master Book: Maths Masti (D. Hurina): write less, cancel more, and keep the working on a single clear line.
Why This Chapter Is Worth More Than It Looks
A direct simplification question is easy to recognise in the paper, and it normally carries one mark. The real value of the chapter is hidden elsewhere. The same skill is used in the data interpretation set, in the ratio questions and in the time, speed and distance section, where a fraction carried incorrectly through three lines spoils an otherwise perfect solution. A candidate who drills simplification finishes the arithmetic portion with time to spare, and that spare time is what allows the reasoning and general knowledge questions to be attempted calmly. In a competitive examination where a handful of marks decides the rank, this quiet chapter is often the difference between a call letter and another year of preparation.
The Order of Operations: Bar, Bracket, Then the Rest
Every simplification obeys one fixed sequence, and the question paper is designed to punish anyone who forgets it. The bar, or vinculum, drawn over a group of terms is the strongest bracket and must be cleared first. After that the brackets are opened from the innermost to the outermost, that is, round first, then curly, then square. Then the powers and the roots are evaluated. Then division and multiplication are carried out from left to right, because the two stand on the same rung of the ladder. Finally addition and subtraction are done, again from left to right, since they too are equal in rank. The examiner knows that a nervous candidate will rush to the addition because it looks the simplest, and this is exactly the trap. A candidate who rewrites the expression on a fresh line after every stage keeps the working honest and prevents a term that has already been used from being counted twice.
Terms You Must Not Confuse
- Expression: a group of numbers joined by the signs of operations.
- Term: a part of the expression that is separated by a plus or a minus sign.
- Bar or vinculum: the line over a group of terms, which overrides everything inside it.
- Numerator: the upper number of a fraction, which counts the parts.
- Denominator: the lower number, which names the size of the parts.
- 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 standing with a proper fraction.
- Reciprocal: the fraction turned upside down, used whenever we divide.
One small word also deserves its own place: the word "of" in a simplification means multiplication, and it is treated with the same priority as a multiplication sign. One fourth of eighty is twenty, and this is worked out before the surrounding division is touched.
Adding and Subtracting Fractions
Fractions can be added or subtracted only when their denominators are the same. The economical way is to find the lowest common multiple of the denominators, which is the smallest number that both divide exactly. Each fraction is then rewritten with that common denominator, and only the numerators are added, while the common denominator stays untouched. At the end the answer is reduced to its lowest terms, which means dividing the numerator and the denominator by whatever factor is common to both. A frequent error is to add the denominators as well, which is wrong because the size of the parts has not changed when fractions are combined. In a police paper question that contains three fractions, it usually pays to bring all three to the common denominator in one step rather than working with two at a time.
Multiplying and Dividing Fractions
Multiplication is the friendliest of the four operations on fractions: multiply the numerators together and the denominators together, and reduce the result. Division is turned into multiplication by flipping the second fraction, that is, by multiplying by its reciprocal. The phrase to remember is that division by a fraction is multiplication by its reciprocal. Before multiplying, it is always worth looking for a numerator that shares a factor with any denominator, because cancelling first keeps the numbers small and makes the final reduction unnecessary. A common slip is to flip the wrong fraction; only the divisor, the fraction that follows the division sign, is turned over, never the first one.
Mixed Numbers and Improper Fractions
A mixed number should be changed into an improper fraction before any operation is attempted. To do this, multiply the whole number by the denominator, add the numerator, and place the result over the same denominator. For example, two and one third becomes seven thirds. This single step prevents a great deal of confusion, because the arithmetic of improper fractions is straightforward, while the arithmetic of mixed numbers invites the mistake of operating on the whole parts and the fractional parts separately. If the question or the options are given as mixed numbers, the final answer may be converted back, but the working is always done in improper form.
Decimals and the Position of the Point
Decimals are best handled by writing them as fractions with denominators that are powers of ten, or by shifting the point until the numbers are whole. When two decimals are added or subtracted, the points must be written one exactly below the other so that the tenths, the hundredths and the thousandths fall in their correct columns. When a decimal is multiplied by a power of ten, the point moves to the right by as many places as there are zeros; when it is divided by a power of ten, the point moves to the left. When one decimal is divided by another, both the divisor and the dividend are multiplied by the same power of ten so that the divisor becomes whole, which leaves the value of the division unchanged. A misplaced point is the single most common reason for a wrong answer in this part of the chapter, and a quick estimate at the end catches it at once.
The Word Of and Complex Fractions
A complex or continuous fraction is one in which a fraction appears in the numerator, in the denominator, or in both. The safe method is to simplify the upper part completely, then simplify the lower part completely, and only then divide the first result by the second. Trying to attack the whole structure in one stroke is where most errors begin. Reading a statement such as the sum of two thirds and one sixth divided by the difference of one half and one third requires the words to be translated into brackets carefully, because the division line itself acts as a large bracket that separates the top from the bottom. Once the expression has been written correctly in one line, the difficulty disappears and what remains is ordinary arithmetic.
Cancellation: the Fastest Tool in the Hall
Cancellation is the greatest time-saver of the chapter. In a product of fractions, a factor 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 small 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. In a timed police paper, ten seconds saved on each sum amount to several extra questions by the end of the arithmetic section.
Powers, Roots and the Word Of Together
Modern police papers sometimes place a power or a square root inside a simplification question. The rule is unchanged: the power or the root is evaluated after the brackets and the word of, but before the division and the multiplication. Thus the square root of the product of two numbers is taken after the product has been formed, while the square of a bracket is taken after the bracket has been simplified. The square roots of perfect squares such as one hundred and forty-four, one hundred and sixty-nine and two hundred and twenty-five are worth remembering, because they appear repeatedly and save precious seconds.
Approximation in the Objective Paper
Since the answer is chosen from four given options, an approximate calculation is often enough, and it is much faster. The numbers are rounded to convenient values, usually to the nearest whole number or to the nearest ten, and the options are then compared. This is safe only when the options are far apart; when two options are very close, the exact working is unavoidable. Even when the exact value is required, a rough estimate made in advance tells the candidate whether the answer is of the order of ten, of a hundred or of a thousand, which immediately exposes a misplaced decimal point or a badly cancelled fraction.
Common Traps in the Police Constable Paper
- Finishing the addition or the subtraction before the division and the multiplication are completed.
- Forgetting to change every sign inside a bracket that is preceded by a minus sign.
- Ignoring the bar over a group of terms, which outranks every bracket.
- Adding fractions without first bringing them to a common denominator.
- Turning over the first fraction instead of the divisor while dividing.
- Operating on a mixed number without first changing it into an improper fraction.
- Losing the position of the decimal point while multiplying or dividing.
- Treating the word of as a weak operation instead of a multiplication that must be done early.
A Seven-Day Revision Plan
- Day one: learn the order of operations until it needs no conscious thought.
- Day two: practise removing nested brackets, giving special attention to the minus sign.
- Day three: drill the addition and subtraction of fractions with unlike denominators.
- Day four: drill the multiplication and division of fractions, cancelling before you multiply.
- Day five: work on mixed numbers, converting them to improper fractions first.
- Day six: practise decimals, taking constant care with the position of the point.
- Day seven: solve timed sets of ten mixed questions and review every wrong step.
Frequently Asked Questions
In what order should a simplification question be solved?
Clear the bar first, then open the brackets from the innermost outwards, then evaluate the powers and roots, then do the division and the multiplication from left to right, and finally the addition and the subtraction from left to right. Division and multiplication rank equally, and so do addition and subtraction, so within each pair the order of appearance decides.
What does the word of mean in a simplification?
It means multiplication, and it is carried out before the ordinary division and multiplication signs. So one third of ninety is thirty, and this is calculated before anything around it is touched. Ignoring this priority is a reliable way to lose the mark.
How do I handle a minus sign before a bracket?
Every sign inside the bracket is reversed when it is opened. Minus, open bracket, a minus b, close bracket becomes minus a plus b. This is the most common mistake in the whole chapter, so slow down for one moment whenever a minus precedes a bracket.
How do I add fractions with different denominators?
Find the lowest common multiple of the denominators, rewrite each fraction with that common denominator, add only the numerators, and reduce. For instance, one half plus one third becomes three sixths plus two sixths, which is five sixths.
Why is it better to cancel before multiplying?
Because it keeps the numbers small, makes the final reduction unnecessary, and reduces the chance of an arithmetic slip. In a product of fractions, any numerator may be cancelled against any denominator, even across two different fractions.