Question

There are a different number of bicycles, tandem bicycles, and tricycles in a shop. There a total of 135 seats, 118 front handle bars, and 269 wheels. How many of each bike is there?

Answer 1

There are 68 bicycles, 17 tandem bicycles, and 33 tricycles in the shop.

Explanation:

Conventionally, a bicycle has a 1 seat, 1 front handle bars and 2 wheels

A tandem bicycle has 2 seats, 1 front handle bar and 2 wheels

A tricycle has 1 seat, 1 front handle bar and 3 wheels

Let's say;

Number of Bicycles = A

Number of Tandem Bicycles = B

Number of Tricycles = C

We can arrive at the equations for each part of the bikes

Seats → 1A + 2B + 1C = 135 -----1

Front handle bars → 1A + 1B + 1C = 118 -------2

Wheels → 2A + 2B + 3C = 269 ------3

Let's manipulate equation 2 to equation 3 by multiplying equation 2 by 2.

We have 2A + 2B + 2C = 236 -------4

Subtracting Eqt 3 from 4, we have

C = 33

Hence, number of tricycles is 33.

Substituting 33 for C in eqts 1 and 2, we have

1A + 2B = 135 - 33 = 102 -------5

1A + 1B = 118 - 33 = 85 -------6

Multiply eqt 6 by 2, we get

2A + 2B = 170 --------7

Subtract eqt 5 from eqt 7, we have

A = 68

Hence, number of bicycles is 68.

Substitute 68 for A in eqt 6, we have

68 + B = 85

B = 85 - 68 = 17

Hence, number of tandem bicycles is 17.

∴ There are 68 bicycles, 17 tandem bicycles, and 33 tricycles in the shop.

To check,

Eqt 1, 1(68) + 2(17) + 1(33) = 135 (Seats)

Eqt2, 68 + 17 + 33 = 118 (Front handle bars)

Eqt3, 2(68) + 2(17) + 3(33) = 269 (Wheels)

Answer 2

Answer: there are 68 bicycles, 17 tandem bicycles and 33 tricycles.

Explanation:

Let x represent the number of bicycles there.

Let y represent the number of tandem bicycles there.

Let z represent the number of tricycles there.

The bicycle and the tricycle has 1 seat each while the tandem bicycle has 2 seats. There a total of 135 seats. This means that

x + 2y + z = 135- - - - - - - - - - - - - - 1

Each of them has 1 front handle bars. The total number of front handle bars is 118. This means that

x + y + z = 118 - - - - - - - - - - - - -2

The bicycle and the tandem bicycle has 2 wheels each while the tricycle has 3 wheels. The total number of wheels is 118. This means that

2x + 2y + 3z = 269 - - - - - - - - - - -3

Subtracting equation 2 from equation 1, it becomes

2y - y = 135 - 118

y = 17

Substituting y = 17 into equation 1 and equation 3, it becomes

x + 17 + z = 118

x + z = 118 - 17

x + z = 101

x = 101 - z

2x + 2 × 17 + 3z = 269

2x + 34 + 3z = 269

2x + 3z = 269 - 34

2x + 3z = 235 - - - - - - - - - - - -3

Substituting x = 101 - z into equation 3, it becomes

2(101 - z) + 3z = 235

202 - 2z + 3z = 235

- 2z + 3z = 235 - 202

z = 33

x = 101 - z = 101 - 33

x = 68