Exercise 1

At a community meeting, there are double the number of women than men and triple the number of children than the total of men and women combined. How many men, women, and children are there if 96 people attend the meeting?

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Exercise 2

A barrel of oil has \frac { 7 }{ 8 } of its content removed. Then, it is filled with 38 liters and the total content is \frac { 3 }{ 5 } the capacity of the barrel. Calculate the total capacity of the barrel.

Exercise 3

A farm has pigs and turkeys. In total, there are 35 heads and 116 paws. How many pigs and turkeys are there on the farm?

Exercise 4

Peter went on a road trip, during which time 20 liters of gasoline were consumed. However, he completed the trip in two stages. In the first, he consumed \frac { 2 }{ 3 } of the gasoline that was in the tank and in the second stage, half of the gasoline that he had left in the tank.

1. Determine the number of liters of gasoline that were in the tank before the trip.

2. Determine the number of liters consumed at each stage.

Exercise 5

In a bookstore, Anne buys a novel with a third of her money and a comic with two-thirds of what remained. When she left the bookstore, she had 12 dollars. How much money did Anne have before arriving at the store?

Exercise 6

A clock sounds to indicate the time of 3 o'clock. At what time between 3 and 4 will the hour and minute needles overlap?

Exercise 7

A clock sounds to indicate the time of 2 o'clock. At what time will its needles form a right angle for the first time?

Exercise 8

A truck leaves a city at a speed of 40 mph. An hour later, a car leaves the same city and travels in the same direction at a speed of 60 mph.

1. How many hours after leaving the city will the car reach the truck?

2. What is the distance from the city where the car will reach the truck?

Exercise 9

At 9 a.m., two cyclists simultaneously leave their houses and travel towards each other on the same road. Houses A and B are located 130 miles away from each other. If the cyclist who leaves from House A pedals at a constant speed of 30 mph, and the cyclist who leaves from House B travels at 20 mph, how far from House A will the two meet and at what time?

Exercise 10

A faucet takes 3 hours to fill a water tank, and another faucet takes four hours to fill a tank of the same size. How long will it take to fill a tank of the same size if both faucets are distributing water together into the same tank?

Exercise 11

A golden brick of 0.950 purity weighs 6300 grams. What amount of pure copper should be added to lower its purity to 0.900?

 

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Solution of exercise 1

At a community meeting, there are double the number of women than men and triple the number of children than the total number of men and women combined. How many men, women, and children are there if 96 people attend the meeting?

Men: x

Women: 2x

Children: 3.(x+2x)=3.3x=9x

x+2x+9x=96

12x=96

x=\frac { 96 }{ 12 }

x=8

 

Men: 8

Women: 2 \times 8 = 16

Children: 9 \times 8 = 72

 

Solution of exercise 2

A barrel of oil has \frac { 7 }{ 8 } of its content removed. Then, it is filled with 38 liters and the total content is \frac { 3 }{ 5 } the capacity of the barrel. Calculate the total capacity of the barrel.

 

Let the total volume of the barrel equal to "x".

The first step is to deduct the \frac { 7 }{ 8 } from the total amount.

x-\frac { 7 }{ 8 } x

\frac { 1 }{ 8 } x

 

The next step is to add the 38 liters which will equal to \frac { 3 }{ 5 } x

\frac { 1 }{ 8 } x+38=\frac { 3 }{ 5 } x

After talking LCM: 5x+1520=24x

1400=19x

\frac { 1400 }{ 19 } =x

x=80

Solution of exercise 3

A farm has pigs and turkeys. In total, there are 35 heads and 116 paws. How many pigs and turkeys are there on the farm?

Let the number of pigs = x

Turkeys = 35 − x

4x+2(35-x)=116

4x+70-2x=116

4x-2x=116-70

2x=46

x=\frac { 46 }{ 2 }

x=23

Pigs = 23

Turkeys = 35 − 23 = 12

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Solution of exercise 4

Peter went on a road trip, during which time 20 liters of gasoline were consumed. However, he completed the trip in two stages. In the first, he consumed \frac { 2 }{ 3 } of the gasoline that was in the tank and in the second stage, half of the gasoline that he had left in the tank.

1. Determine the number of liters of gasoline that were in the tank before the trip.

First stage. \frac { 2 }{ 3 } x

Second stage. \frac { 1 }{ 2 } (x-\frac { 2 }{ 3 } x)=\frac { 1 }{ 2 } .\frac { 1 }{ 3 } x=\frac { 1 }{ 6 } x

\frac { 2 }{ 3 } x+\frac { 1 }{ 6 } x=20

4x+x=120

5x=120

x=\frac { 120 }{ 5 }

x=24

2. Determine the number of liters consumed at each stage.

First stage: \frac { 2 }{ 3 } \times 24=16

Second stage: \frac { 1 }{ 6 } \times 24=4

Solution of exercise 5

In a bookstore, Anne buys a novel with a third of her money and a comic with two-thirds of what remained. When she left the bookstore, she had 12 dollars. How much money did Anne have before arriving at the store?

Let the total amount that Anne had equal to "x"

Cost of novel = \frac { 1 }{ 3 } x

Cost of comic = \frac { 2 }{ 3 } (x-\frac { 1 }{ 3 } x)=\frac { 2 }{ 3 } .\frac { 2 }{ 3 } x=\frac { 4 }{ 9 } x

The remainder cost = 12

 

\frac { 1 }{ 3 } x+\frac { 4 }{ 9 } x+12=x

3x+4x+108=9x

3x+4x-9x=-108

-2x=-108

2x=108

x=\frac { 108 }{ 2 }

x=54 dollars

Solution of exercise 6

A clock sounds to indicate the time of 3 o'clock. At what time between 3 and 4 will the hour and minute needles overlap?

Keep in mind that the angle or arc representing the minute hand's position is always 12 times greater than the arc that describes the hour hand.

x is the arch that describes the needle hourly.

(15 + x) is the arch that describes the minute hand.

15 + x = 12x

15 = 12x - x

15 = 11x

x=\frac { 15 }{ 11 }

The needles will overlap at 3:16 and 21 seconds.

 

Solution of exercise 7

A clock sounds to indicate the time of 2 o'clock. At what time will its needles form a right angle for the first time?

Turning clockwise, the needles will form a right angle approximately at 2:25. Therefore, let x be the arc that describes the hour needle.

x is the arc that describes the hour needle.

25 + x, is the arc that describes the minute hand.

25 + x = 12x

25 = 12x - x

25 = 11x

x=\frac { 25 }{ 11 }

The clock will form a 90° angle at 2:27 and 16 seconds.

 

Solution of exercise 8

A truck leaves a city at a speed of 40 mph. An hour later, a car leaves the same city and travels in the same direction at a speed of 60 mph.

1. How many hours after leaving the city will the car reach the truck?

 

Speed = \frac { Distance }{ Time }

Distance = Speed \times Time

 

Distance traveled by car = { d }_{ 1 } = 60\times (t-1) = 60(t-1)

Distance traveled by truck = { d }_{ 2 } = 40 \times t = 40t

{ d }_{ 1 } = { d }_{ 2 }

60(t-1) = 40t

60t - 60 = 40t

60t - 40t = 60

20t = 60

t = \frac { 60 }{ 20 }

t = 3

 

As the car leaves the city one hour later than the truck, the time it will take to reach the truck will be (3-1)=2 hours.

 

2. What is the distance from the city where the car will reach the truck?

{ d }_{ 2 } = 40t

{ d }_{ 2 } = 40(3)

{ d }_{ 2 } = 120 miles

Solution of exercise 9

At 9 a.m., two cyclists simultaneously leave their houses and travel towards each other on the same road. Houses A and B are located 130 miles away from each other. If the cyclist who leaves from House A pedals at a constant speed of 30 mph, and the cyclist who leaves from House B travels at 20 mph, how far from House A will the two meet and at what time?

The cyclist travels at 30 mph from house A and his friend travels at 20 mph from house B. They will meet up at a certain place and we will call that place C. If we add the distance of AC and CB, we know that their sum will be 130 (because that is the overall distance). To find those distances, we will use speed formula since we know the speed and we will assume that at "t", they will meet each other.

 

Speed = \frac { Distance }{ Time }

Distance = Speed \times Time

 

Distance traveled by the cyclist from house A = { d }_{ 1 } = 30 \times t = 30t

Distance traveled by the cyclist from house B = { d }_{ 2 } = 20 \times t = 20t

 

{ d }_{ 1 } + { d }_{ 2 } = 130

30t + 20t = 130

50t = 130

t = \frac { 130 }{ 50 }

t = 2.6 hours

 

Let's convert 2.6 hours into hour:min

0.6 \times 60 = 36 minutes

time = 2 hour and 36 minutes

They will meet at 11:36 am.

 

{ d }_{ AC } = 30 \times \frac { 130 }{ 50 }

{ d }_{ AC } = 78 miles

Solution of exercise 10

A faucet takes 3 hours to fill a water tank, and another faucet takes four hours to fill a tank of the same size. How long will it take to fill a tank of the same size if both faucets are distributing water together into the same tank?

 

In one hour the first faucet fills \frac { 1 }{ 3 } of the tank.

In one hour the second faucet fills \frac { 1 }{ 4 } of the tank.

 

In one hour two together faucets will have filled:

\frac { 1 }{ 3 } + \frac { 1 }{ 4 } = \frac { 1 }{ x }

\frac { 4 + 3 }{ 12 } = \frac { 1 }{ x }

\frac { 7 }{ 12 } = \frac { 1 }{ x }

7x = 12

x = \frac { 7 }{ 12 } hours

 

Solution of exercise 11

A golden brick of 0.950 purity weighs 6300 grams. What amount of pure copper should be added to lower its purity to 0.900?

Gold Copper Total
No. of g 6300 x 6300 + x
Pure Gold 0.950 \times 6300 0.900 \times (6300 + x)

0.900 \times (6,300 + x) = 0.950 \times 6300

5670 + 0.900x = 5985

0.900x = 315

x = \frac { 315 }{ 0.900 } = 350

Copper 350 grams

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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.