Efficient Evaluation of Integer and Fractional Expressions

Hello! This is the first lesson in Numerical Fluency and Percentage Foundations, the opening module of your CAT quantitative-aptitude course. Before tackling percentages and arithmetic word problems, you need a reliable way to evaluate expressions without losing time to sign errors, unnecessary multiplication, or ambiguous order of operations.

Today you will work with integers and fractions, using four connected ideas: follow the structure of the expression, determine signs, cancel common factors, and factor before calculating when it helps. The aim is not merely to get an answer, but to get it accurately with less arithmetic.

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1. Read the expression before calculating

An expression tells you which quantities belong together. Brackets make that grouping explicit; a fraction bar groups its entire numerator and its entire denominator. Within a group, evaluate powers if present, then multiplication and division, then addition and subtraction.

Multiplication and division have equal priority: in an ungrouped chain, work from left to right. The same is true of addition and subtraction. For example,

$$24÷6\times 2=(24÷6)\times 2=8,$$

not $24÷(6\times 2)=2$. The latter is a different expression because it introduces brackets that were not there.

The following video makes this distinction particularly clear.

A CAT-useful distinction: Left-to-right evaluation protects the meaning of an ungrouped expression. Once you have rewritten a product correctly as factors, you may choose an easier calculation order. Thus $12\times 6÷2=12\times (6/2)=36$. But $18÷3\times 4$ cannot be changed to $18÷(3\times 4)$: that moves the $4$ into a denominator where it did not belong.

A long fraction bar also acts as grouping. In

$$\frac{18-6\times 2}{3+3},$$

evaluate the top and bottom separately before dividing. The numerator is $6$, the denominator is $6$, and the value is $1$.

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2. Settle the sign first

When multiplying or dividing, two negative factors give a positive result; one negative factor gives a negative result. With several factors, count the negative ones: an even number gives a positive product, and an odd number gives a negative product.

For addition and subtraction, it is often clearer to rewrite subtraction as addition of a negative number. Consider

$$-12-(-7)+3(-4).$$

The subtraction of $-7$ becomes addition of $7$, while $3(-4)=-12$. Hence

$$-12+7-12=-17.$$

Writing the sign change explicitly is faster than having to diagnose a wrong answer later. Also keep a leading negative sign attached to its term when regrouping: $15-8+2$ is $15+(-8)+2=9$, not $15-(8+2)$.

If powers appear, brackets decide whether a negative sign is part of the base: $(-3{)}^2=9$, whereas $-3^2=-(3^2)=-9$. We will study exponent laws later; for now, this distinction prevents a common sign error.

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3. Cancel factors, not terms

Fractions become much easier when you simplify before multiplying. A fraction can be reduced because dividing its numerator and denominator by the same nonzero factor leaves its value unchanged. The word factor matters: it means a quantity joined by multiplication, not a term joined by addition.

Read OpenStax’s explanation of that principle and its multiplication examples before applying it to the worked calculation below.

The image below shows both kinds of cancellation you may use in a product: between the numerator and denominator of one fraction, or across two fractions. Both work because all the numerators form one product and all the denominators form another.

For instance, in $4/15\times 5/8$, divide $4$ and $8$ by $4$, then $5$ and $15$ by $5$. Only $1/3\times 1/2$ remains.

Division of fractions has one additional step: replace division by a nonzero fraction with multiplication by its reciprocal. The reciprocal keeps the sign and swaps numerator with denominator; zero has no reciprocal.

Now combine sign, reciprocal, and cancellation:

$$-\frac{9}{14}÷\frac{3}{7}=-\frac{9}{14}\times \frac{7}{3}.$$

Cancel $9$ with $3$ to leave $3$ and $1$; cancel $7$ with $14$ to leave $1$ and $2$. The answer is $-3/2$, without ever multiplying $9\times 7$.

Addition is different. To calculate $-5/6+7/9$, use a common denominator:

$$-\frac{5}{6}+\frac{7}{9}=-\frac{15}{18}+\frac{14}{18}=-\frac{1}{18}.$$

You cannot cancel a number from one term of a sum against a denominator elsewhere. Cancellation applies to factors of an entire numerator and denominator.

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4. Factor before doing large arithmetic

Factorization makes a hidden common multiplier visible. Compare the two ways to evaluate

$$\frac{84\times 25-84\times 10}{15\times 28}.$$

Multiplying both products first works, but creates avoidable numbers. Both numerator terms contain $84$, so factor it out:

$$\frac{84(25-10)}{15\times 28}=\frac{84\times 15}{15\times 28}=3.$$

This is a legitimate cancellation because the whole numerator is now a product. By contrast, in $(12+6)/6$, cancelling the $6$ from just one term would be invalid. You may first factor the numerator as $6(2+1)$; only then can the common $6$ be cancelled, giving $3$.

Here is the complete approach in a mixed expression:

$$E=\frac{(-18)\times 35}{(-21)\times 15}-\left(\frac{3}{4}÷\frac{9}{8}\right).$$

The first fraction is positive because it has two negative factors. Cancelling $18$ with $21$, and $35$ with $15$, reduces it to $2$. In the brackets, division by $9/8$ means multiplication by $8/9$, which reduces to $2/3$. Therefore

$$E=2-\frac{2}{3}=\frac{4}{3}.$$

Notice the order of decisions: identify the grouped parts and sign, convert division if necessary, cancel factors, then combine the remaining terms. For CAT speed, this is more dependable than multiplying everything immediately and simplifying a large final fraction.

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5. A short fluency routine

For a few minutes after this lesson, revisit two worked examples in the OpenStax reading with their solutions covered. Before doing any arithmetic, mark the grouped parts, predict the sign, and identify possible cancellations. Then compare your steps with the printed solutions. If you make an error, record whether it came from precedence, sign, reciprocal, or invalid cancellation; that label is more useful for revision than simply marking an answer wrong.

The module’s later timed set will use CAT previous-year questions. At this stage, prioritize a method you can execute correctly every time; speed will follow as these decisions become familiar.

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Key takeaways

Read grouping before calculating. Treat multiplication and division as equal-priority operations, determine signs explicitly, and cancel only common factors in a numerator and denominator. When a numerator contains a sum or difference, factor it first if you want to cancel a shared multiplier.

In the next lesson, you will move between fractional, decimal, and percentage forms—representations that occur throughout CAT arithmetic.