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Check the solution
?
\latex{ 5 }
\latex{ 7 }
\latex{ 6 }
\latex{ 237 }
\latex{ 237 }
¢
¢
Example 1
Hugo fired six arrows at the target shown in the image. Which of the following could be his score if he hit the target each time?
\latex{ A: 4 } \latex{ B: 17 } \latex{ C: 56 } \latex{ D: 28 } \latex{ E: 31 }
\latex{ 1 }
\latex{ 3 }
\latex{ 5 }
\latex{ 7 }
\latex{ 9 }
Solution
- As all six arrows hit the target, Hugo scored at least six points; thus, solution \latex{ A } is wrong.
- The most points can be scored if each arrow hits the ring, which is worth \latex{ 9 } points: \latex{6\times9=54}, therefore, solution \latex{ C } is also wrong.
- The score cannot be \latex{ 17 } or \latex{ 31 } (solutions \latex{ B } and \latex{ E }) either, because each ring is worth an odd number of points. Therefore, the sum of the six scores can only be an even number.
- Solution \latex{ D } is left, which is a possible outcome.
For example, \latex{7+5+5+5+5+1=28} or \latex{9+7+7+3+1+1=28}.
Answer:
Solution \latex{ D } is a possible result.
Incorrect solutions can easily be identified if you calculate with the last digits of the numbers, determine whether the result is an odd or even number, or estimate the value of the solution.
Example 2
A two-digit number is divisible by five and the sum of its digits is \latex{ 7 }. What is this number?
Solution
The number is divisible by five; thus, its last digit is either \latex{ 0 } or \latex{ 5 }.
The sum of its digits is seven.
If the last digit is \latex{ 0 }, then the tens digit is \latex{ 7 }. In this case, the two-digit number is \latex{ 70 }.
If the last digit is \latex{ 5 }, then the tens digit is \latex{ 2 }. In this case, the two-digit number is \latex{ 25 }.
If the last digit is \latex{ 0 }, then the tens digit is \latex{ 7 }. In this case, the two-digit number is \latex{ 70 }.
If the last digit is \latex{ 5 }, then the tens digit is \latex{ 2 }. In this case, the two-digit number is \latex{ 25 }.
Check:
Both \latex{ 70 } and \latex{ 25 } are divisible by \latex{ 5 }.
The sum of the digits is \latex{ 7 + 0 = 7 } and \latex{ 2 + 5 = 7 }.
Both requirements are fulfilled.
Both \latex{ 70 } and \latex{ 25 } are divisible by \latex{ 5 }.
The sum of the digits is \latex{ 7 + 0 = 7 } and \latex{ 2 + 5 = 7 }.
Both requirements are fulfilled.
Answer:
Two numbers meet the requirements: \latex{ 70 } and \latex{ 25 }.
Exercises can have several solutions, so you have to make sure that you have found all of them.
Example 3
There are three plates with sandwiches on them. There are twice as many sandwiches on the second plate as on the first plate, and three times as many sandwiches on the third plate as on the first plate.
If you move \latex{ 22 } sandwiches from the third plate to the first plate, then there will be \latex{ 15 } more sandwiches on the first plate than on the second plate. How many sandwiches will be on the first plate?
Solution
Illustrate the number of sandwiches on the first plate with a line segment.
plate \latex{ 1 }:
plate \latex{ 2 }:
plate \latex{ 3 }:
If you move \latex{ 22 } sandwiches from the third plate to the first one, then there will be \latex{ 15 } more sandwiches on the first plate than on the second one. Illustrate it with a diagram.
plate \latex{ 1 }:
plate \latex{ 2 }:
\latex{ 22 }
\latex{ 15 }
The red line segment, which represents \latex{ 22 - 15 = 7 } sandwiches, is equal to the line segment that illustrates the initial number of sandwiches on the first plate. Therefore, there were originally \latex{7} sandwiches on the first plate, \latex{ 14 } sandwiches on the second plate and \latex{ 21 } sandwiches on the third plate.
Check:
There are twice as many sandwiches on the second plate as on the first one: \latex{ 14 = 2 ×7 }, and three times as many sandwiches on the third plate as on the first one: \latex{ 21 = 3 × 7 }. However, you cannot move \latex{ 22 } sandwiches from the third plate to the first plate, as there are only \latex{ 21 } on it. This is a contradiction.
Answer:
The exercise has no solution.
It is possible that an exercise does not have a solution.
Before answering the question of a problem, check whether your solution is correct. There are different ways to do this.
- Check whether your result fulfils the requirements described in the text.
- Make estimations to rule out incorrect solutions.
- Examine the divisibility and other properties (even or odd, positive or negative, fraction or integer, etc.) of your result.
- Make sure that you have found all the possible solutions.
- Do not forget that some exercises have no solutions.
Exercises
{{exercise_number}}. Gabe solved three word problems and wrote down the solutions; however, he forgot which result belonged to which exercise. Match the results with the exercises.
- The Smith family spends €\latex{3 } on milk and bread every \latex{ day. } How much do they spend on milk and bread over a \latex{ week? }
- Mr. Smith bought \latex{ 30\,kg } of potatoes for €\latex{ 2 \,per\,kg}. How much did he pay?
- Mr. Smith had to pay €\latex{ 38 } for a hair dryer. How much money did he get back if he paid with a €\latex{ 200 } note?
- €\latex{ 162 }
- €\latex{ 21 }
- €\latex{ 60 }
{{exercise_number}}. Did Monica or Rachel have more money in her wallet
- if they had the same amount of money after Monica received €\latex{ 2 };
- if they had the same amount of money after Monica bought Rachel an ice cream?
{{exercise_number}}. George has eight cats. When the cats are extremely tired, they purr for exactly three minutes before falling asleep. When the cats are moderately tired, they purr for exactly five minutes before falling asleep. And when the cats are only a little tired, they purr for exactly seven minutes before falling asleep. On Tuesday, all of George's cats fell asleep at the same time. Decide whether the following statements are true or false.
- George's cats purred for at least \latex{ 24 } \latex{ minutes } on Tuesday.
- George's cats purred for exactly one \latex{ hour } on Tuesday.
- George's cats purred for a maximum of \latex{ 56 } \latex{ minutes } on Tuesday.
- George's cats purred for an odd number of \latex{ minutes } on Tuesday.
{{exercise_number}}. Decide whether the following statements are true or false.
- The sum of two odd numbers is always an even number.
- The sum of three odd numbers is always an odd number.
- The sum of an even and an odd number can be an even number.
- The sum of two even numbers cannot be an odd number.
{{exercise_number}}. Decide whether the following statements are true or false.
- The product of two odd numbers can be an even number.
- The product of an even and an odd number is always an even number.
- The half of every even number is an odd number.
- The product of every number that is multiplied by two is an even number.
{{exercise_number}}. How many numbers can the \latex{\textcolor{c7dcf0} \blacksquare} represent in the multiplication \latex{367,83} \latex{\textcolor{c7dcf0} \blacksquare} \latex{\times\; 4,563} if you know that the product is an odd number?
{{exercise_number}}. Pete says, 'I have thought of a number and added the number one greater to it. The last digit of the sum is two. What number did I think of?'
Kate thinks it is impossible. Is she right? Why or why not?
{{exercise_number}}. Louis says, 'I have thought of a number. I multiplied it by itself and then added the number to the product. The last digit of the sum is \latex{ 5 }. What number did I think of?'
Eve thinks it is impossible. Is she right? Why or why not?
{{exercise_number}}. What numbers can be in the hundreds place in the addition shown in the image?
\latex{ 4 }
\latex{ 2 }
\latex{ + }
\latex{ 0 }
\latex{ 1 }
{{exercise_number}}. The sum of three numbers is \latex{ 624 }. The second number is equal to the first one, and the third number is \latex{ 36 } greater than the second one. How many of the three addends are even numbers?
{{exercise_number}}. The teacher spilt some ink on her notebook during a school trip, and now her calculations of the costs are smudged. Can you tell what numbers are covered in ink if she wrote them under each other according to place values?
a)
b)
Each of the \latex{20} plane tickets costs ...
I paid
Each of the \latex{18} hotel rooms costs ...
I paid
altogether.
altogether.
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
€
€
€
€
\latex{ 9 }
\latex{ 6 }
\latex{ 2 }
\latex{ 2 }
\latex{ 7 }
{{exercise_number}}. There are books on three bookshelves. There are twelve more books on the second bookshelf than on the first one, and half as many books on the third bookshelf as on the first one.
If you move \latex{ 40 } books from the second bookshelf to the third, then there will be \latex{ 27 } more books on the third bookshelf than on the first one. Then, how many books will be on the first bookshelf?
{{exercise_number}}. Cut a square of \latex{ 8 × 8 } units into smaller pieces and then reassemble them into a rectangle, as shown in the image. The area of the original square is \latex{ 64 } units, while that of the resulting rectangle is \latex{ 7 × 9 = 63 } units.
One square unit is lost. How is this possible?
Quiz
There are six apples on a plate. Can you divide them equally among six children so that one apple remains on the plate?