Practice Reasoning vs Computation in Math

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

Worksheet PDF Free Answer-key PDF Family

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

\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

7^{4}

, reason with the cycle of last digits

7,9,3,1

. Give it.

Example 3

medium

Without using a calculator, determine: is

\ln(e^5 \cdot e^3)

best computed as

\ln(e^8)

or by adding logs? Give the value.

Example 4

medium

To find

\frac{2}{3}

\frac{3}{2}

, reason that they are reciprocals. Give the product.

Example 5

easy

Without computing, determine whether

997 \times 1003

is greater than, less than, or equal to

1000^2

. Then verify by computation.

Example 6

easy

999999\times0

better solved by multiplying or by reasoning? Give the value.

Example 7

challenge

Which is bigger,

2^{60}

3^{40}

? Reason by writing both as

(\cdot)^{20}

and give the larger base.

Example 8

hard

Reason: how many digits does

2^{10} \cdot 5^{10}

have?

Example 9

hard

Reason:

x + \frac{1}{x} = 3

. Find

x^2 + \frac{1}{x^2}

without solving for

x

Example 10

medium

Without computing

25!

, determine the highest power of

5

that divides

25!

Example 11

hard

Without computing each, find

\gcd(2^{100}, 2^{75})

Example 12

challenge

Reason: a fair coin is flipped

10

times. By symmetry, what is

P(\text{more heads than tails}) + P(\text{more tails than heads})

Example 13

challenge

4\times4

grid: how many squares of all sizes? Reason with

\sum k^2

rather than counting one by one.

Example 14

medium

Without expanding, find the coefficient of

x^2

(x+1)(x+2)(x+3)

Example 15

medium

To compute

98\times102

, reason via

(100-2)(100+2)

. Give the product.

Example 16

easy

Reason (don't divide): is

3{,}456{,}789

divisible by

3

? Use the digit-sum rule. Give

1

for yes,

0

for no.

Example 17

easy

\sqrt{2}+\sqrt{2}

better found by decimals or by reasoning? Give the exact value.

Example 18

medium

Reason: if

f(x) = x^3

and

g(x) = x^{1/3}

, what is

f(g(8))

without composing symbolically?

Example 19

medium

\frac{0}{5}

a computation or a definition issue? Give the value.

Example 20

challenge

To evaluate

\sum_{k=1}^{99}\frac{1}{k(k+1)}

, reason with telescoping (

\frac1k-\frac1{k+1}

). Give the value.

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Related Concepts

Proof (Intuition)Explanation vs Derivation