Practice Recomposition in Math

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

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Quick Recap

Recomposition is the process of combining simpler parts, sub-results, or solved sub-problems back together to form a complete solution or to understand the whole structure from its pieces.

After decomposing a problem, you must reassemble the pieces correctly — like completing a jigsaw puzzle, the boundary conditions between parts must match.

Showing a random 20 of 50 problems.

Example 1

medium

Two probabilities found:

P(A) = 0.4

P(B) = 0.5

with

A,B

independent. Recombine for

P(A \cap B)

Example 2

easy

You split

48 = 50 - 2

to multiply

48 \times 7

. Recompose to give the product.

Example 3

easy

Partial fractions gave

\frac{1}{x-1}+\frac{1}{x+1}

. Recombine into a single fraction.

Example 4

easy

Partial fractions yielded

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

. Recombine into a single fraction.

Example 5

challenge

A combinatorial count is split: pick a chair (

5

ways) then assign people (

3! = 6

ways). Recombine for the total arrangements.

Example 6

medium

Solving a system in two pieces, you found

x = 4

from elimination, then needed

y

. Substituting into

3x + 2y = 16

, recombine for

y

Example 7

medium

Vector components

\langle 3,0\rangle

and

\langle0,4\rangle

were found. Recombine into one vector and give its magnitude.

Example 8

challenge

Solving a system, you found

x=2

from one equation and need

y

. Given

2x+y=7

, recombine to finish, then verify in

x-y=-1

Example 9

easy

Distance pieces:

4\text{ km north}

then

3\text{ km east}

. Recompose for straight-line distance from start.

Example 10

easy

Times broken into legs:

1.5\text{ h} + 2.25\text{ h}

. Recombine for total time.

Example 11

easy

Two rooms have area

24\text{ ft}^2

and

36\text{ ft}^2

. Recombine for total area.

Example 12

medium

A region's area was split as the integral

\int_0^1 x\,dx

minus

\int_0^1 x^2\,dx

. Recombine to a number.

Example 13

hard

Complex number

z = 3 + 4i

was split into real and imaginary parts. Recombine to find

z\bar z

Example 14

medium

Modular sub-results

x\equiv2\pmod3

and

x\equiv3\pmod5

were found. Recombine via CRT for the smallest positive

x

Example 15

medium

Two overlapping sets have

|A|=10

|B|=8

|A\cap B|=3

. Recombine for

|A\cup B|

Example 16

hard

Polynomial factored to

(x-2)(x^2 + 3x + 5)

. Recompose and identify the coefficient of

x

Example 17

easy

Two trip legs took

2

h and

1

h. Recombine for total time.

Example 18

challenge

After decomposing

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

, recombine into

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

Example 19

hard

A solid is decomposed: a cylinder of volume

40\pi

minus a hemisphere of volume

\frac{16}{3}\pi

scooped out. Recombine for the remaining volume.

Example 20

medium

Cases

x=8

and

x=-2

solve an absolute-value equation. Recombine into the solution set.

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Related Concepts

Decomposition