Recomposition Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Recomposition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Recomposition reassembles solved subproblems into the whole answer, making sure the seams between parts match.

Common stuck point: The procedure for recomposition is the easy part; the trap is assuming recomposition is just adding. Asking "Have I already solved the pieces, and is my job now to fit them into one whole with matching seams?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Have I already solved the pieces, and is my job now to fit them into one whole with matching seams?

Worked Examples

Example 1

easy

After decomposing

x^2 + 5x + 6 = (x+2)(x+3)

, recompose to solve

x^2+5x+6 = 0

Answer

x = - 2 or x = - 3

First step

Decomposition gave:

x^2+5x+6 = (x+2)(x+3)

Full solution

2
Set the product equal to zero: $(x+2)(x+3) = 0$ .
3
Recompose the solution: either $x+2=0$ (giving $x=-2$ ) or $x+3=0$ (giving $x=-3$ ).
4
Final answer: $x \in \{-2, -3\}$ .

Recomposition uses the pieces obtained in decomposition to assemble the final answer. The zero-product property lets us read off solutions directly from the factored form.

Example 2

medium

Partial fractions gave

\frac{5x+1}{(x+1)(x-2)} = \frac{4}{3(x+1)}+\frac{11}{3(x-2)}

. Recompose to verify by adding the fractions.

Example 3

medium

A trapezoid is decomposed into two triangles via a diagonal. The triangles have area

15

and

9

. Recompose the trapezoid's area.

Example 4

medium

\sum_{k=1}^{4} k = 10

and

\sum_{k=1}^{4} k^2 = 30

. Recombine for

\sum_{k=1}^{4} k(k+1)

Example 5

medium

Found legs

a = 7

and

b = 24

of a right triangle. Recompose for the hypotenuse.

Example 6

hard

Complex number

z = 3 + 4i

was split into real and imaginary parts. Recombine to find

z\bar z

Example 7

hard

Modular pieces

n \equiv 1 \pmod 4

n \equiv 2 \pmod 3

. Recombine via CRT for the smallest positive

n

Example 8

challenge

A combinatorial count is split: pick a chair (

5

ways) then assign people (

3! = 6

ways). Recombine for the total arrangements.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

You find

x=3

and

y=1

from solving a system. Recompose by substituting back into both equations to verify.

Example 2

medium

Given the identity

\sin^2\theta = \frac{1-\cos 2\theta}{2}

, recompose: use it to evaluate

\int \sin^2\theta\, d\theta

Example 3

easy

You found two rectangle areas

12

and

8

that make an L-shape. What is the total area?

Example 4

easy

Partial fractions gave

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

. Recombine into a single fraction.

Example 5

easy

You computed

7\times10=70

and

7\times2=14

. Recombine for

7\times12

Example 6

easy

Two trip legs took

2

h and

1

h. Recombine for total time.

Example 7

easy

You found roots

u=2

and

u=3

for

u=\sqrt x

. Recombine to get the

x

-solutions.

Example 8

easy

Place-value pieces

100+20+3

recombine to what number?

Example 9

easy

A cylinder volume is

\pi r^2 h

and a removed cone is

\frac13\pi r^2 h

. Recombine (subtract) for the remaining volume.

Example 10

easy

You split

\int(2x+\cos x)dx

into

x^2

and

\sin x

. Recombine the antiderivative.

Example 11

medium

Two overlapping sets have

|A|=10

|B|=8

|A\cap B|=3

. Recombine for

|A\cup B|

Example 12

medium

Sub-sums

\sum k^2=\frac{n(n+1)(2n+1)}{6}

and

\sum k=\frac{n(n+1)}{2}

were found. Recombine for

\sum (k^2+k)

n=3

Example 13

medium

A complex number was split as

z=3+4i

. Recombine to find its modulus

|z|

Example 14

medium

Vector components

\langle 3,0\rangle

and

\langle0,4\rangle

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

Example 15

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 16

medium

Modular sub-results

x\equiv2\pmod3

and

x\equiv3\pmod5

were found. Recombine via CRT for the smallest positive

x

Example 17

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 18

challenge

Pieces of a piecewise function:

f(x)=x^2

for

x\le1

and

f(x)=2x-1

for

x>1

. Recombine and check continuity at

x=1

Example 19

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 20

medium

Cases

x=8

and

x=-2

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

Example 21

medium

Digit-stage counts

9,9,8

were found for distinct-digit

3

-digit numbers. Recombine for the total.

Example 22

medium

A trapezoid was split into a rectangle (area

bh

) and a triangle (area

\frac{(a-b)h}{2}

, with

a>b

). Recombine into one formula.

Example 23

easy

After factoring

x^2 - 9 = (x-3)(x+3)

, recompose to solve

x^2 - 9 = 0

Example 24

easy

Distance pieces:

4\text{ km north}

then

3\text{ km east}

. Recompose for straight-line distance from start.

Example 25

easy

Partial fractions yielded

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

. Recombine into a single fraction.

Example 26

easy

Sub-results

\sin\theta = 3/5

and

\cos\theta = 4/5

. Recombine for

\tan\theta

Example 27

medium

After splitting

\int (3x^2 + 4x)\,dx

into

\int 3x^2\,dx

and

\int 4x\,dx

, recombine the antiderivative.

Example 28

medium

Vector components

\langle 5, 12\rangle

were found from a problem. Recombine into the vector's magnitude.

Example 29

medium

Sub-counts:

4

shirts and

3

pants. Recombine for the number of outfits (one of each).

Example 30

medium

Two probabilities found:

P(A) = 0.4

P(B) = 0.5

with

A,B

independent. Recombine for

P(A \cap B)

Example 31

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 32

hard

A region between

y=x^2

and

y=4

was split into left and right halves by symmetry, each with area

\frac{16}{3}

. Recompose for the total area.

Example 33

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 34

hard

Solving

|2x - 1| = 5

, two cases gave

x = 3

and

x = -2

. Recompose the solution set and verify.

Example 35

hard

After computing

f'(x) = 2x

from

f(x) = x^2 + 5

, recompose by integrating

f'

to recover

f

(up to a constant).

Example 36

challenge

A piecewise function:

f(x) = x

for

x \le 0

f(x) = x^2

for

x > 0

. Recompose: is

f

continuous at

0

Example 37

challenge

Eigenvalues of a

2\times2

matrix are

\lambda_1 = 3, \lambda_2 = 5

. Recombine for the determinant and trace.

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Related Concepts

Decomposition

Background Knowledge

These ideas may be useful before you work through the harder examples.

decomposition meta