Twój koszyk jest pusty
Working backwards
\latex{10 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 6 }
\latex{ 7 }
\latex{8 }
\latex{9 }
\latex{5 }
\latex{4 }
Milo the mouse followed a sketch map to find his way to a piece of cheese. At each crossroad, he decided which path to follow. Determine the point where Milo began his journey if he took the following directions at the crossroads.
\latex{ N }
\latex{ S }
\latex{ E }
\latex{ N }
\latex{ W }
\latex{ N }
\latex{ E }
\latex{ E }
\latex{ E }
\latex{ E }
\latex{ S }
It is easier to solve certain problems by starting at the end and working backwards.
Example 1
I have thought of a number. If you divide it by five, then add six to the quotient, then multiply the sum by eight, you get \latex{ 80 }. What number did I think of?
Solution
Follow the changes of the original number.
\latex{ 1 }st number
\latex{ 2 }nd number
\latex{ 3 }rd number
\latex{ 4 }th number
\latex{5\times4=20}
\latex{10-6=4}
\latex{80\div 8=10}
\latex{ 80 }
\latex{\times8}
\latex{+6}
\latex{\div5}
the number
I have thought of
\latex{\div8}
\latex{-6}
\latex{\times5}
the number
you got
you got
The original number is \latex{ 20 }.
Check:
\latex{20\div 5=4}; \latex{4+6=10}; \latex{10\times8=80}, which is in accordance with the text of the exercise.
Answer:
Therefore, the number I have thought of was \latex{ 20 }.
Example 2
There are three trees in front of our house: a peach tree, a walnut tree and a cherry tree. One morning, a total of \latex{ 48 } sparrows perched on the three trees. Later in the day, eight sparrows flew from the peach tree to the walnut tree, and six sparrows flew from the walnut tree to the cherry tree. After these changes, the number of sparrows on each tree was equal. How many sparrows were on each tree in the morning?
Solution
After the changes, an equal number of sparrows sat on each tree; thus, \latex{48\div 3=16} sparrows were on one tree. Fill in the table with the number of sparrows on each tree throughout the day.
peach tree
walnut tree
cherry tree
\latex{16}
\latex{16-6=10}
\latex{\color{e50051}{10}}
\latex{16}
\latex{16}
after the changes
during the day
\latex{16+6=22}
in the morning
\latex{\color{e50051}{-6}}
\latex{\color{e50051}{+6}}
\latex{\color{e50051}{-8}}
\latex{\color{e50051}{+8}}
\latex{16+8=\color{e50051}{24}}
\latex{22-8=\color{e50051}{14}}
\latex{16}
\latex{\color{0095db}{-8}}
\latex{\color{0095db}{+8}}
\latex{\color{0095db}{-6}}
\latex{\color{0095db}{+6}}
Check:
After the changes, \latex{ 24 - 8 = 16 } sparrows were on the peach tree, \latex{ 14 + 8 - 6 = 16 } were on the walnut tree, and \latex{ 10 + 6 = 16 } were on the cherry tree.
Answer:
According to the table, in the morning, \latex{ 24 } sparrows perched on the peach tree, \latex{ 14 } on the walnut tree, and \latex{ 10 } on the cherry tree.
Example 3
A bowl was filled with dumplings. Ben came home first and ate half of the dumplings plus half of one more dumpling. Then Aaron arrived and ate half of the remaining dumplings, leaving five dumplings in the bowl. How many dumplings were in the bowl originally?
Solution
Represent the number of dumplings by a line segment.
half of all the dumplings
Ben ate this many dumplings
Aaron ate this many dumplings
half of the remaining
dumplings
this is also half of
the remaining dumplings
\latex{ 5 }
\latex{\frac{1}{2}}
Aaron ate half of the dumplings that Ben left. The other half of these dumplings were the remaining five, which means that Aaron ate five dumplings as well. Therefore, Ben left \latex{ 2 × 5 = 10 } dumplings. If Ben had not eaten half of one more dumpling, he would have eaten exactly half of the total number of dumplings, which is 1\latex{0\frac{1}{2}} .
Therefore, there were \latex{10\frac{1}{2}\times2=21} dumplings in the bowl originally.
Check:
There were \latex{ 21 } dumplings. Ben ate \latex{21\div 2+\frac{1}{2}=10\frac{1}{2}+\frac{1}{2}=11} dumplings. Ten dumplings were left.
Aaron ate \latex{10\div 2=5} dumplings.
There were \latex{ 21 } dumplings. Ben ate \latex{21\div 2+\frac{1}{2}=10\frac{1}{2}+\frac{1}{2}=11} dumplings. Ten dumplings were left.
Aaron ate \latex{10\div 2=5} dumplings.
As a result, five dumplings remained.
Answer:
There were 21 dumplings in the bowl originally.
Exercises
{{exercise_number}}. I have thought of a number. If you subtract \latex{ 29 } from it, then multiply the difference by \latex{ 17 }, and then divide the product by \latex{ 221 }, you get four. What number did I think of?
{{exercise_number}}. I have thought of a number. If you add \latex{ 38 } to it, then divide the sum by \latex{ 10 }, then multiply the quotient by nine, then add \latex{ 19 } to the product, you get \latex{ 100 }. What number did I think of?
{{exercise_number}}. \latex{ 25 } is five less than half of a number. What is this number?
{{exercise_number}}. After performing a series of operations, Pete got \latex{ 520 } as a result. Later, he realised that in his last operation, he added \latex{ 89 } instead of subtracting \latex{ 89 }. What is the correct result?
{{exercise_number}}. After performing a series of operations, Paul got \latex{ 480 } as a result. Later, he realised that in his last operation, he multiplied by four instead of dividing by four. What is the correct result?
{{exercise_number}}. Andrew, Ben and Clark were playing a game. According to the rules, the winner of each round received five coins from each of the other players. At the end of the sixth round, each boy had \latex{ 60 } coins. Ben won the sixth, Clark won the fifth, and Andrew won the fourth round. How many coins did each of the boys have at the end of the third round?
{{exercise_number}}. A swarm of bees flew into our garden. Half of the bees landed on the peach tree, half of the rest of the bees landed on the goldenrod flowers, and the remaining \latex{ 18 } bees landed on the tulips. How many bees flew into our garden?
{{exercise_number}}. Rachel and Aaron took a lot of pictures during their holiday in Paris. On Wednesday, half of their photos were taken at the Eiffel Tower, two-thirds of the rest of their photographs were taken at the Notre Dame, and the remaining eight pictures were taken at the Arc de Triomphe. How many photos did they take on Wednesday?
{{exercise_number}}. A shop first sold \latex{ 5 \;metres }, then \latex{ 3 \;metres }, and then \latex{ 4.5 \;metres } of a roll of fabric. Later, a seamstress bought half of the remaining fabric. Then another seamstress purchased \latex{ 10 } more \latex{ metres } of it. The last customer bought the rest of the fabric, which was \latex{ 2 \;metres } long. How long was the roll of fabric?
{{exercise_number}}. If you double the length of the opposite sides of a rectangle and triple the length of its other opposite sides, you get a rectangle whose perimeter is \latex{ 48\;cm }. How long are the sides of the original rectangle?
{{exercise_number}}. One-third of the students in a class are girls. One-fourth of the boys in the class play basketball. There are \latex{ 12 } boys in the class who do not play basketball. How many students are there in the class?
*{{exercise_number}}. On Monday, a used car dealer sold half of his cars plus half of one more car. On Tuesday, he sold half of the remaining cars plus half of one more car. On Wednesday, he sold half of the rest of his cars plus half of one more car, which left him with only one car in his dealership. The dealer did not buy any new cars during the week. How many cars did he sell on Monday?
*{{exercise_number}}. A wealthy man passed away and left half of his \latex{ gold\, coins }, plus \latex{ 1,000 } more, to his wife. His daughter inherited half of the remaining \latex{ coins } plus \latex{ 1,000 } more. His butler received half of the \latex{ coins } that were left, plus \latex{ 1,000 } more. The man left half of the rest of the \latex{ coins }, plus \latex{ 1,000 } more, to his dog. Then, the remaining \latex{ 10,000 } \latex{ gold\, coins } were donated to charity. How many \latex{ gold\, coins } did the wealthy man have?
Game
Two players play this game with one piece. The piece begins at the START. The players take turns moving the piece one to five spaces forward at a time. The player who enters the FINISH wins the game.
How can the first player guarantee their win?
FINISH
START
\latex{ 48}
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
\latex{ 11 }
\latex{ 12 }
\latex{ 13 }
\latex{ 14 }
\latex{ 15 }
\latex{ 16 }
\latex{ 17 }
\latex{ 18 }
\latex{ 19 }
\latex{ 20 }
\latex{ 21 }
\latex{ 22 }
\latex{ 23 }
\latex{ 24 }
\latex{ 25 }
\latex{ 26 }
\latex{ 27 }
\latex{ 28 }
\latex{ 29 }
\latex{ 30 }
\latex{ 31 }
\latex{ 32 }
\latex{ 33 }
\latex{ 34 }
\latex{ 35 }
\latex{ 36 }
\latex{ 37}
\latex{ 38}
\latex{ 39}
\latex{ 40}
\latex{ 41}
\latex{ 42}
\latex{ 43}
\latex{ 44}
\latex{ 45}
\latex{ 46}
\latex{ 47}
Quiz
There are \latex{ 30\; litres } of oil in a barrel. How can you measure out exactly \latex{ 6\; litres } of oil without wasting a single drop, using only a \latex{ 4 }-\latex{ litre } bottle and a \latex{ 9 }-\latex{ litre } bottle?