Practice Reasoning vs Computation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Reasoning is the process of understanding why a mathematical fact is true and how ideas connect, while computation is the mechanical process of calculating an answer — both are essential but serve different purposes.

Computation is following a recipe; reasoning is deciding which recipe to use and why. Most math mistakes come from computing when you should be reasoning first.

Showing a random 20 of 50 problems.

Example 1

medium
Reason: is 1421+721\frac{14}{21} + \frac{7}{21} better done by adding or by recognizing the result? Give the value.

Example 2

medium
To find the last digit of 747^{4}, reason with the cycle of last digits 7,9,3,17,9,3,1. Give it.

Example 3

medium
Without using a calculator, determine: is ln⁡(e5⋅e3)\ln(e^5 \cdot e^3) best computed as ln⁡(e8)\ln(e^8) or by adding logs? Give the value.

Example 4

medium
To find 23\frac{2}{3} of 32\frac{3}{2}, reason that they are reciprocals. Give the product.

Example 5

easy
Without computing, determine whether 997×1003997 \times 1003 is greater than, less than, or equal to 100021000^2. Then verify by computation.

Example 6

easy
Is 999999×0999999\times0 better solved by multiplying or by reasoning? Give the value.

Example 7

challenge
Which is bigger, 2602^{60} or 3403^{40}? Reason by writing both as (⋅)20(\cdot)^{20} and give the larger base.

Example 8

hard
Reason: how many digits does 210⋅5102^{10} \cdot 5^{10} have?

Example 9

hard
Reason: x+1x=3x + \frac{1}{x} = 3. Find x2+1x2x^2 + \frac{1}{x^2} without solving for xx.

Example 10

medium
Without computing 25!25!, determine the highest power of 55 that divides 25!25!.

Example 11

hard
Without computing each, find gcd⁡(2100,275)\gcd(2^{100}, 2^{75}).

Example 12

challenge
Reason: a fair coin is flipped 1010 times. By symmetry, what is P(more heads than tails)+P(more tails than heads)P(\text{more heads than tails}) + P(\text{more tails than heads})?

Example 13

challenge
A 4×44\times4 grid: how many squares of all sizes? Reason with ∑k2\sum k^2 rather than counting one by one.

Example 14

medium
Without expanding, find the coefficient of x2x^2 in (x+1)(x+2)(x+3)(x+1)(x+2)(x+3).

Example 15

medium
To compute 98×10298\times102, reason via (100−2)(100+2)(100-2)(100+2). Give the product.

Example 16

easy
Reason (don't divide): is 3,456,7893{,}456{,}789 divisible by 33? Use the digit-sum rule. Give 11 for yes, 00 for no.

Example 17

easy
Is 2+2\sqrt{2}+\sqrt{2} better found by decimals or by reasoning? Give the exact value.

Example 18

medium
Reason: if f(x)=x3f(x) = x^3 and g(x)=x1/3g(x) = x^{1/3}, what is f(g(8))f(g(8)) without composing symbolically?

Example 19

medium
Is 05\frac{0}{5} a computation or a definition issue? Give the value.

Example 20

challenge
To evaluate ∑k=1991k(k+1)\sum_{k=1}^{99}\frac{1}{k(k+1)}, reason with telescoping (1k−1k+1\frac1k-\frac1{k+1}). Give the value.
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