Exercise 1

A beacon flashes its light every 12 seconds, another every 18 seconds, and a third every minute. At 6.30 pm the three flash simultaneously.

Find out the times when the three flash simultaneously again in the next five minutes.

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Exercise 2

A businessman goes to Chicago every 18 days for one day and another businessman every 24 days, also for only one day. Today, both men are in Chicago.

Within how many days will the two businessmen be in Chicago again at the same time?

Exercise 3

What is the smallest number that when divided on separate occasions, 15, 20, 36, and 48, gives the remainder of 9 in every case?

Exercise 4

There are 3 differently sized casks of wine in a cellar whose capacities are: 250 liters, 360 liters, and 540 liters. The owner of the cellar wants to package the wine in barrels with an equal amount of wine in each one. Calculate the maximum capacities of these barrels so that the owner can package equal amounts of wine in each cask, and determine the quantity of barrels he will need.

Exercise 5

The floor of a room that needs to be tiled is 5 m long and 3 m wide.

Determine the ideal size of the tiles and the number of the tiles needed, such that the number of tiles that are placed is minimal and none of them are to be cut. Keep in mind that all tiles are to be the same size.

Exercise 6

A trader wants to put 12,028 apples and 12,772 oranges into boxes. Each box is to contains an equal number of apples and an equal number of oranges and also the greatest number of each. Find the ideal number of oranges and apples for each box and the number of boxes needed.

Exercise 7

What is the size of the largest possible square tile that can fit an in a room 8 m long and 6.4 meters wide without being cut? How many tiles are needed?

 

 

Solution of exercise 1

A beacon flashes its light every 12 seconds, another every 18 seconds, and a third every minute. At 6.30 pm the three flash simultaneously.

Find out the times when the three flash simultaneously again in the next five minutes.

12 = 2 \times 2 \times 3 = { 2 }^{ 2 } \ times 3

18 = 2 \times 3 \times 3 = 2 \times { 3 }^{ 2 }

60 = 2 \times 2 \times 3 \times 5 = { 2 }^{ 2 } \times 3 \times 5

LCM (12, 18, 60) = { 2 }^{ 2 } \times { 3 }^{ 2 } \times 5 = 180

180:60 = 3

 

Solution of exercise 2

A businessman goes to Chicago every 18 days for one day and another businessman every 24 days, also for only one day. Today, both men are in Chicago.

Within how many days will the two businessmen be in Chicago again at the same time?

18 = 2 \times { 3 }^{ 2 }

24 = { 2 }^{ 3 } \times 3

LCM (18, 24) = { 2 }^{ 3 } \times { 3 }^{ 2 } = 72

Within 72 days.

 

Solution of exercise 3

What is the smallest number that when divided on separate occasions, 15, 20, 36, and 48, gives the remainder of 9 in every case?

LCM (15, 20, 36, 48) = { 2 }^{ 4 } \times { 3 }^{ 2 } \times 5 = 720

720 + 9 = 729

 

Solution of exercise 4

There are 3 differently sized casks of wine in a cellar whose capacities are: 250 liters, 360 liters, and 540 liters. The owner of the cellar wants to package the wine in barrels with an equal amount of wine in each one. Calculate the maximum capacities of these barrels so that the owner can package equal amounts of wine in each cask, and determine the quantity of barrels he will need.

GCD (250, 360, 540) = 10

The capacity of the barrels= 10 liters

Number of barrels of C1 = \frac{250}{10} = 25

Number of barrels of C2 = \frac{360}{10} = 36

Number of barrels of C3 = \frac{540}{10} = 54

Number of barrels = 25 + 36 + 54 = 115 barrels.

 

Solution of exercise 5

The floor of a room that needs to be tiled is 5 m long and 3 m wide.

Determine the ideal size of the tiles and the number of the tiles needed, such that the number of tiles that are placed is minimal and none of them are to be cut. Keep in mind that all tiles are to be the same size.

3 m = 30 dm \qquad 30 = 2 \times 3 \times 5

5 m = 50 dm \qquad 50 = 2 \times { 5 }^{ 2 }

A = 30 \times 50 = 1,500 dm²

G.C.D. (30, 50) = 2 \times 5= 10 dm aside

{ A }_{ b }= { 10 }^{ 2 } = 100 dm 2

1,500 dm² : 100 dm² = 15 tiles

 

Solution of exercise 6

A trader wants to put 12,028 apples and 12,772 oranges into boxes. Each box is to contains an equal number of apples and an equal number of oranges and also the greatest number of each. Find the ideal number of oranges and apples for each box and the number of boxes needed.

GCD (12,028, 12,772) = 124

124 oranges in each box.

Boxes of oranges = \frac{ 12,772 }{ 124 } = 103

Boxes of apples = \frac { 12,028 }{ 124 } = 9712,028/124 = 97

Boxes necessary = 103 + 97 = 200

 

Solution of exercise 7

What is the size of the largest possible square tile that can fit an in a room 8 m long and 6.4 meters wide without being cut? How many tiles are needed?

8 m = 80 dm \qquad 80 = { 2 }^{ 4 } \times 5

6.4 m = 64 dm \qquad 64 = { 2 }^{ 6 }

GCD (80, 64) = { 2 }^{ 4 } = 16 dm aside

{ A }_{ b } = { 16 }^{ 2 } = 256dm²

A = 80 \times 64 = 5,120 dm²

5,120 dm² : 256 dm² = 20 tiles

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.