Сепетиңиз бош
Draw a diagram
The treasure is located \latex{ 15 \;feet } from one-third of half the distance between the palm tree and the cave.
Example 1
Fifteen seamen serve on a pirate ship. How many sailors and officers are among them if there are four times as many sailors as officers?
Solution
Represent the number of officers by a line segment:
Four times that line segment represents the number of sailors:
The sum of the two is equal to five times the line segment:
officers
sailors
The diagram shows that the number of officers is one-fifth of that of the seamen, i.e., \latex{ 15÷5=3 }. The number of sailors is four times as many; therefore, \latex{ 4×3 = 12 }.
Check:
The number of seamen serving on the pirate ship is \latex{ 3 + 12 = 15 }.
The number of sailors is \latex{ 4 × 3 = 12 }. Both requirements are fulfilled.
The number of sailors is \latex{ 4 × 3 = 12 }. Both requirements are fulfilled.
Answer:
There are three officers and twelve sailors on the pirate ship.
Example 2
Come up with different exercises that can be represented by the following diagrams.
Nick:
Fred:
Bill:
Tom:
a)
b)
\latex{ 20 }
Solution
- For example, Nick has \latex{ 20 } fewer gold coins than Fred.
Fred has \latex{ 20 } more gold coins than Nick.
If Nick gets \latex{ 20 } more gold coins, he will have the same number of gold coins as Fred.
If Fred spends \latex{ 20 } gold coins, he will have the same number of gold coins as Nick.
- Bill has one-fifth as many gold coins as Tom.
Tom has five times as many gold coins as Bill.
If you multiply the number of gold coins Bill has by five, he will have as many gold coins as Tom.
If you divide the number of gold coins Tom has by five, he will have as many gold coins as Bill.
Example 3
You break a \latex{ 46 \;cm } long skewer into two pieces. One of the pieces is \latex{ 14 \;cm } longer than the other. How long is each piece?
Solution 1
Draw a diagram
shorter:
longer:
\latex{ 14 }
longer
shorter
shorter
sum
\latex{ 14 }
Two times the length of the shorter piece is \latex{ 14 \;cm } less than the sum of the lengths of the two pieces: \latex{ 46 - 14 = 32\;(cm) }. Therefore, the shorter piece is \latex{ 32÷2 = 16 \;cm } long, while the longer one is \latex{ 16 + 14 = 30 \;cm } long.
Check:
The sum of the lengths of the two pieces is \latex{ 16 + 30 = 46 \;cm }.
Answer:
One of the pieces is \latex{ 16\; cm } and the other is \latex{ 30\; cm } long.
Solution 2
shorter:
longer:
\latex{ 14 }
longer
shorter
longer
sum
\latex{ 14 }
Two times the longer piece is \latex{ 14 \;cm } longer than the sum of the lengths of the two pieces: \latex{ 46 + 14 = 60 \;(cm) }. Therefore, the longer piece is \latex{ 60 ÷ 2 = 30 \;cm } long, while the shorter one is \latex{ 30 - 14 = 16 \;cm } long.
You get the same answer as in Solution 1.
Answer:
One of the pieces is \latex{ 16 \;cm } and the other is \latex{ 30 \;cm } long.
Example 4
A boat has covered half the distance between two islands. If it sails five more \latex{ miles }, only one-third of the distance will remain. How far are the two islands from each other?
Solution
Represent the distance between the two islands by a line segment. To be able to mark both the half and one-third of the distance, divide the line segment into six equal parts, as six is the smallest positive integer that is divisible by both \latex{ 2 } and \latex{ 3 }.
The diagram shows that one-sixth of the distance is \latex{ 5 \;miles }; thus, the total distance between the two islands is \latex{ 6 × 5 = 30 \;miles }.
\latex{ 5 }
one-third
of the distance
of the distance
half of the distance
Check:
Half of the distance is \latex{ 30÷2 = 15 \;miles }. If the boat sails five more \latex{ miles }, then \latex{ 20 \;miles } of the journey will be covered. This means that \latex{ 30 - 20 = 10 \;miles } will be left, which is exactly one-third of the total distance. The requirements are fulfilled.
Answer:
The distance between the two islands is \latex{ 30 \;miles }.
Exercises
{{exercise_number}}. Match the texts with the corresponding diagrams.
- Kate is four \latex{ years } older than Peter. The sum of their ages is \latex{ 20 }.
Kate:
Peter:
Together:
Peter
Peter
Peter
Kate
Kate
Kate
\latex{ 4 }
\latex{ 4 }
\latex{ 4 }
\latex{ 4 }
\latex{ 20 }
\latex{ 20 }
\latex{ 20 }
A
B
C
- Mark's dad is six times as old as his son. The sum of their ages is \latex{ 35 }.
Mark:
Dad:
Together:
Dad
M
\latex{ 6 }
\latex{ 35 }
\latex{ 35 }
\latex{ 35 }
D
E
F
Dad
Dad
M
M
{{exercise_number}}. Andrew and Ben collect cards. They have a total of \latex{ 50 } cards together. Which of the following texts can be represented by the diagram?
- Andrew has eight fewer cards than Ben.
- If Ben got eight more cards, he would have the same number of cards as Andrew.
- Andrew has twice as many cards as Ben.
- Andrew has eight more cards than Ben.
Andrew:
Ben:
\latex{ 8 }
- Andrew has four times as many cards as Ben.
- Ben has five more cards than Andrew.
- The number of cards Andrew has is one-fourth of the number of cards Ben has.
- The number of cards Andrew has is one-fifth of the number of cards Ben has.
Andrew:
Ben:
Draw a diagram for each of the following exercises and solve them with their help
{{exercise_number}}. A tower with a dome is \latex{ 138 \;m } tall. The dome is \latex{ 102 \;m } shorter than the tower without the dome. How tall is the dome?
{{exercise_number}}. A violin with a case costs €\latex{ 3,200 }. The case costs €\latex{ 2,500 } less than the violin. How much does the violin cost without the case?
{{exercise_number}}. A bottle with a cork costs \latex{ 110 \;cents }. The bottle is \latex{ 100 \;cents } more expensive than the cork. How much does the cork and the bottle cost separately?
{{exercise_number}}. If you decrease the lengths of the opposite sides of a rectangle by \latex{ 5\; cm } each, you get a square. What is the perimeter of the resulting square if the perimeter of the rectangle is \latex{ 54\; cm ?}
{{exercise_number}}. I have thought of two numbers. Their difference is \latex{ 435 }, and their sum is \latex{ 819 }. What are the two numbers?
{{exercise_number}}. Tim is twelve \latex{ years } old. In how many \latex{ years } will he be three times as old as he is now?
{{exercise_number}}. What time is it if twice as much time has passed since noon as there is left until midnight?
{{exercise_number}}. Tom dug a \latex{ 1 \;m \;40 \;cm } deep hole in the ground. While standing in the hole, Ian came by and asked how much deeper he planned to dig. Tom replied that if he digs \latex{ 80 \;cm } more, then his head will be below ground level just as much as it is above ground level now. How tall is Tom?
{{exercise_number}}. I have thought of a number. If you add four times the number to three times the number, you get \latex{ 77 }. What number did I think of?
{{exercise_number}}. A right angle is divided into two angles by a ray. What are the sizes of the angles if half of one of the angles is equal to one-third of the other angle?
{{exercise_number}}. Ella and Eli are two baby elephants. Ella is \latex{ 180 \;kg } heavier than Eli, making her three times as heavy as Eli. How much does Eli weigh?
{{exercise_number}}. There are three sisters: Jasmine, Jane and Joanne. The sum of their ages is \latex{ 21 }. Jane is twice as old as Joanne, and Jasmine is twice as old as Jane. How old are the sisters?
{{exercise_number}}. Kate was travelling from London to Liverpool by bus. She fell asleep exactly halfway there. When she woke up, the distance to Liverpool was exactly half the distance the bus had covered while Kate was asleep. What portion of the travel did Kate spend sleeping?
{{exercise_number}}. If you divide a number by one-fifth of itself, you get five. What is this number?
{{exercise_number}}. Half the number of horses in a stable is five more than one-fourth of the number of horses. How many horses are in the stable?
{{exercise_number}}. A mountain climber at the last rest stop said, 'If I complete two-thirds of the climb and then climb an additional \latex{ 70\; metres }, there will be \latex{ 40\; metres } less than one-fourth of the total distance to the summit left.' How far is the climber from the summit?
{{exercise_number}}. How old is the person who will be twice as old five years from now as he was five years ago?
{{exercise_number}}. Pam said to he friend, 'If I had five more As, I would have one and a half times as many As as if I had five less As.' How many As does Pam have?
{{exercise_number}}. Arnold read one-third of his new book. If he reads \latex{ 18 } more pages, the number of pages left will be \latex{ 42 } pages more than what he has already read. How many pages is the book?
{{exercise_number}}. I thought of a number. I added two to it, then multiplied the sum by two, and then subtracted two times a number that is two less than the original number from the product. What is the resulting number?
{{exercise_number}}. Peter and Paul live \latex{ 25 \;km } from each other. They start cycling towards each other at the same time. When they meet, they realise that Peter has cycled seven \latex{ km } more than half the distance that Paul has covered. How many \latex{ km } did Paul cycle?
Quiz
It takes \latex{ 90 \;minutes } to drive from city \latex{ A } to city \latex{ B }. Two cars start travelling towards each other from the two cities at the same time. One of the cars travels half as fast as the other. They meet after \latex{ 60 \;minutes }.
Which car is further from city \latex{ A } when they meet?
