Practice Decomposition in Math

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

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Quick Recap

The strategy of breaking a complex mathematical object or problem into simpler, independent subproblems that can be solved separately.

Divide and conquer: a hard problem of size $n$ becomes $n$ easy problems. Long division, partial fractions, and integration by parts all use decomposition.

Showing a random 20 of 50 problems.

Example 1

medium

Decompose the proof that

\sqrt{2} + \sqrt{3}

is irrational into logical sub-goals.

Example 2

easy

Decompose solving the system

2x + y = 7

and

x - y = 2

into steps.

Example 3

easy

To find the area of a shape made of a

4\times 5

rectangle on top of a

6\times 2

rectangle, what is a natural decomposition?

Example 4

hard

Decompose a

7 \times 8

chessboard count of dominoes (

1\times 2

tiles). Approximate using a known result.

Example 5

medium

Decompose

108

into prime factors and use it to find the number of divisors.

Example 6

medium

Decompose the area between

y=x

and

y=x^2

[0,1]

into the right subproblem.

Example 7

easy

Decompose

\frac{5}{6}

as a sum of unit fractions.

Example 8

easy

Decompose

\frac{3}{4}

as a sum of unit-related parts to compute

\frac{3}{4}

20

Example 9

challenge

Decompose

\sum_{k=1}^{n} k(k+1)

into simpler known sums and evaluate for general

n

Example 10

challenge

Decompose the proof that

n^3-n

is divisible by

6

for all integers

n

Example 11

medium

Decompose

x^3 - 8

as a product of factors.

Example 12

hard

Decompose

\int \frac{1}{x^2-1}\,dx

via partial fractions and evaluate.

Example 13

easy

Decompose the count: how many ways to pick an outfit from

3

shirts and

4

pants?

Example 14

easy

Decompose

24

into prime factors.

Example 15

medium

Decompose

\cos(2\theta)

into a function of

\cos\theta

only.

Example 16

medium

Decompose the area of a regular hexagon (side

s

) into triangles.

Example 17

hard

Decompose the proof that the product of two odd numbers is odd.

Example 18

easy

To compute

99 \times 7

, decompose the

99

Example 19

easy

To find the area of an L-shaped region, what is a natural decomposition?

Example 20

medium

Decompose solving

|2x - 6| = 8

into cases.

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Related Concepts

Recomposition