Estimating Numerical Expressions and Checking Plausibility

Hello! In the last lesson, you simplified powers and roots without expanding everything. You also used a quick check to see whether one answer made sense. This lesson makes that check systematic: you will estimate expressions, judge whether an answer is plausible, and decide when an estimate is precise enough for a CAT Quantitative Aptitude question.

An estimate is not a guess. It is a calculation with deliberately simpler numbers. The skill lies in choosing numbers that are easy to handle and noticing how much your choices might affect the result.

1. Find the scale before the exact value

Suppose a calculation gives $2.52$ for $2.1\times 1.2$. Because the factors are near $2$ and $1$, a result near $2$ makes sense. A result of $25.2$ would signal a misplaced decimal point before you revisited the multiplication.

That is an order-of-magnitude check: is the answer around $2$, $20$, or $200$? It is often the fastest way to catch an error. For a closer estimate, retain more of the original numbers. For example,

Here, convenient numbers give more useful precision than rounding both terms to the nearest ten.

Watch the opening of How to Estimate in Maths from Cognito. The worked fraction shows how to simplify each part of an expression before judging the result.

In that example, the estimated numerator is $20\times 7=140$, and the estimated denominator is $190-40=150$. Their quotient is $14/15$, which is near $1$. This is enough to reject an answer such as $10$, though it might not distinguish two options that are both close to $1$.

Choose the precision to match the decision. If you need only the scale, round aggressively. If nearby values matter, preserve another digit or use a better calculation.

2. Choose compatible numbers, then inspect your rounding

Compatible numbers are nearby numbers selected because they work well together. The image below changes $23\times 67$ into the mentally convenient product $25\times 70=1,750$.

That result tells you the product is in the thousands, not the hundreds or tens of thousands. But both factors went up, so the estimate must exceed the exact product. If you need a closer estimate, leave one factor unchanged:

You now know that the true product is below $1,610$. In fact, the earlier compatible-number estimate is too high to distinguish tightly spaced answer choices; convenience alone does not guarantee closeness.

For fractions, compatibility may mean choosing numbers that simplify to a familiar ratio. The mathcentre numeracy booklet demonstrates this with $\frac{51}{73}$ of $300$: replace $\frac{51}{73}$ by $\frac{50}{75}=\frac{2}{3}$, giving an estimate of $200$.

There is more information in that approximation than just “about $200$.” The estimate is low: $51$ is greater than $50$, while $73$ is less than $75$, so the original positive fraction is greater than $\frac{50}{75}$. The calculated value, about $209.59$, lies on the expected side.

For positive numbers, recording the direction of rounding makes an estimate a stronger check. If both factors of a product increase, its estimate increases. For a positive quotient, increasing the numerator raises it, while increasing the denominator lowers it. When one change pushes up and another pushes down, you cannot infer the net direction without looking more closely.

3. Use bounds when “about” is not enough

An estimate gives a useful central value; a bound gives a range the answer cannot escape. Bounds are especially effective for rejecting an answer that looks superficially reasonable.

Because $65<67<70$, multiplying throughout by positive $23$ gives

Thus the product lies between $1,495$ and $1,610$. You need not calculate $23\times 67$ to rule out $1,750$ as an exact answer.

The same idea checks division. Since $0.49<0.5$ and both numbers are positive,

An answer of $40$ is impossible. This catches a common decimal-place mistake: dividing by a number near one-half should produce roughly twice the numerator, not one-fifth of it.

Bounds also build naturally on the previous lesson's roots. Since $14^2=196$ and $15^2=225$,

Consequently, $\sqrt{205}+3.9$ lies between $17.9$ and $18.9$. An answer of $22$ fails the check even if you do not approximate the square root any further.

Beware subtraction of nearly equal numbers. Rounding $100.2-99.8$ to $100-100$ gives zero, although the actual difference is $0.4$. The rounding changes are large relative to the answer, despite being small relative to either starting number. When two terms nearly cancel, retain the digits where they differ. Likewise, before using directional comparisons, check signs: the simple product and quotient rules above assume positive numbers.

4. Match the estimate to the CAT task

Estimation is most valuable when it changes what you need to calculate. In a multiple-choice question with widely separated options, a rough result may settle the choice. With close options, the first estimate is a starting point, not permission to stop.

Watch the selected segment of Speed Maths 6: Approximation with Options from Rodha. It concentrates on this decision: how the spacing of the answer choices affects the precision you need.

Consider

A first pass gives

If the options were $10,51,103,1003$, the scale alone would make $103$ the viable choice. But if they were $99,101,103,105$, “about $100$” would not be enough. Refine the part that is still uncertain:

Then $5.06\times 20.3\approx 102.7$, so $103$ fits the closer options. You did not need to carry out long division to many decimal places; you needed just enough precision to separate the candidates.

This suggests a reliable test-day habit: estimate, inspect the choices, then refine only if needed. For a question without answer choices, an estimate remains an error check, but it does not replace an exact answer when the question asks for one.

Key takeaways

Estimate with numbers that preserve the expression's scale and are easy to calculate. Note whether your rounding pushes a result high or low; use bounds when you need a firm plausibility test. Be especially careful with division by numbers below $1$ and with subtraction of nearly equal terms. In CAT multiple-choice questions, let the spacing of the options determine whether a rough estimate is sufficient or a more precise calculation is worth the time.

Next, you will translate verbal comparisons into algebraic expressions and equations. The same plausibility habit will help you check whether the quantities and relationships in your model make sense.