Сепетиңиз бош
What is the question?
How much more is the number of your eyes than the number of the eyes of a regular person?
The number of your eyes is how much more than two?
How many more eyes do I need to have as many as you have?
If you cannot answer a question immediately, then you are faced with a problem that needs to be solved. To do so, you first have to understand the text and the question.
Example 1
At Gabe's birthday party, his two aunts asked him all sorts of questions. Gabe noticed that although they phrased the questions differently, they both asked the same things.
Aunt Thea's questions were the following:
- How many \latex{ years } ago were you born?
- How much did you grow in the past two \latex{ years? }
- What is the age difference between you and your younger brother?
- How many \latex{ years } older is your father than you?
Aunt Mal's questions were the following:
- How much shorter were you two \latex{ years } ago?
- How old was your father when you were born?
- How old are you now?
- Compared to you, how much younger is your brother?
Match the questions to which the same answers can be given.
Solution
Gabe's age is equal to the number of \latex{ years } that have passed since he was born; thus, questions \latex{ (a) } and \latex{ (C) } have the same answer.
Gabe has grown as much in the past two \latex{ years } as he was shorter two \latex{ years } ago; thus, \latex{ (b) } – \latex{ (A) }.
The age difference between Gabe and his younger brother is equal to the number of years his brother is younger than him; thus, \latex{ (c) } – \latex{ (D) }.
The number of years Gabe's dad is older than Gabe is the same as his age when Gabe was born; thus, \latex{ (d) } – \latex{ (B) }.
Example 2
The sum of the lengths of two sides of a rectangle that meet at a vertex is \latex{ 48 \;cm }.
Which of the following questions
- can be answered without calculation;
- can be answered after performing a calculation;
- cannot be answered?
\latex{ a }
\latex{ b }
\latex{a+b=48\;cm}
- What is the perimeter of the rectangle?
- How long are the sides of the rectangle?
- What is the area of the rectangle?
- How much longer is the larger than the shorter side?
- What is the sum of the lengths of the adjacent sides of the rectangle?
Solution
- The perimeter of a rectangle is equal to the sum of the lengths of its sides. Since the opposite sides of a rectangle are equal in length, its perimeter can be calculated by multiplying the sum of the lengths of two adjacent sides by two: \latex{2\times 48=96\;(cm)}.
Therefore, the perimeter of the rectangle is \latex{ 96 \,cm }.
This question can only be answered after performing a calculation.
- The lengths of the sides of a rectangle cannot be determined based on the sum of the lengths of the sides that meet at a vertex.
If the lengths of the sides of the rectangle are whole numbers, then the shorter side can be any whole \latex{ centimetre } from \latex{ 1\;cm } to \latex{ 24 \;cm }. The length of the longer side is also a whole \latex{ centimetre }. In this case, \latex{ 24 } different rectangles satisfy the conditions.
However, if the lengths of the sides of the rectangle can be fractions as well, then an infinite number of different rectangles satisfy the conditions (for example, the shorter side is \latex{ 0.2\;cm }, and the longer side is \latex{ 47.8\;cm }).
- In exercise \latex{ b) }, it was concluded that there are several different rectangles whose sides meeting at a vertex have a sum of \latex{ 48\;cm }. As the areas of these rectangles are different, it is impossible to determine the area of the rectangle.
The table shows that the rectangle, whose perimeter has a constant value, has the largest area when its sides are equal in length, i.e., when the rectangle is a square.
\latex{ a\\(cm) }
\latex{ b \\(cm) }
\latex{ A\\(cm^{2})}
\latex{ 1 }
\latex{ 2 }
\latex{\vdots}
\latex{\vdots}
\latex{\vdots}
\latex{\vdots}
\latex{\vdots}
\latex{\vdots}
\latex{\vdots}
\latex{ 3 }
\latex{ 4 }
\latex{ 44 }
\latex{ 176 }
\latex{ 135 }
\latex{ 45 }
\latex{ 46 }
\latex{ 47 }
\latex{ 47 }
\latex{ 92 }
\latex{ 23 }
\latex{ 575 }
\latex{ 25 }
\latex{ 576 }
\latex{ 24 }
\latex{ 24 }
\latex{ 12 }
\latex{ 36 }
\latex{ 432 }
- The sum of the lengths of the two sides of a rectangle that meet at a vertex is known \latex{ (a+b=48\,cm) }. Based on this information alone, the difference between the lengths of the sides cannot be determined.
- Two sides that meet at a vertex are called adjacent sides. The sum of the lengths of the adjacent sides of the rectangle is \latex{ 48 \;cm. } This question can be answered without calculation.
Exercises
{{exercise_number}}. Two classes in the sixth grade were playing against each other in the school's basketball championship. After the game, Kate and Helen asked Zoe about the results. Zoe soon realised that although the girls phrased their questions differently, they were asking the same things. Match the questions to which the same answers can be given.
Kate's questions were the following:
- Who won?
- What was the point difference between the two basketball teams?
- How many substitutions were made during the game?
- Did you really score more points than all the other players in your team combined?
Helen's questions were the following:
- Is it true that you scored more than half of the points your team scored?
- How many more points did your team score than the other team?
- In addition to the starting lineup, how many more students played in the game?
- Which team scored more points?
{{exercise_number}}. Gabe is \latex{ 11 } \latex{ years } old and \latex{ 156\;cm } tall. When he was \latex{ 9 } \latex{ years } old, he was \latex{ 138\;cm } tall. His younger brother, Pete, is now as old as Gabe was two \latex{ years } ago. Pete was \latex{ 122\;cm } tall then, but now he is \latex{ 139\;cm }. Their father is \latex{ 38 } \latex{ years } old.
Which of the following questions can be answered without calculation; can be answered after performing a calculation; cannot be answered?
- How old is Gabe’s brother?
- How tall is their father?
- Who is taller now, Gabe or Peter?
- Who grew more in the past two \latex{ years }, Peter or Gabe?
- How old was Gabe's father when Pete was born?
{{exercise_number}}. The product of the lengths of the adjacent sides of a square is \latex{ 81 } \latex{dm^{2}}.
Which of the following questions can be answered without calculation; can be answered after performing a calculation; cannot be answered?
- What is the area of the square?
- How long are the sides of the square?
- What is the perimeter of the square?
- How much longer is one of the sides of the square than the side adjacent to it?
{{exercise_number}}. A straight angle is divided into four equal angles.
Which of the following questions can be answered without calculation; can be answered after performing a calculation; cannot be answered?
- What is the size of the resulting angles?
- What is the sum of the resulting angles?
- What fraction of the straight angle is one resulting angle?
- What is the size of an angle that is three-fourths of the straight angle?
{{exercise_number}}. Ben and Zack are brothers, and they have a dog called Chips. Every morning, Ben takes Chips on a \latex{ 15 }-\latex{ minute } walk, while Zack takes the dog on a \latex{ 30 }-\latex{ minute } walk every evening. Read the questions below and match them to the correct answers marked with capital letters.
- \latex{30 \div 15 = \frac{30}{15} = 2}
- \latex{15\div 30=\frac{15}{30}=\frac{1}{2}}
- \latex{\left( 30+15 \right)\div 15=\frac{45}{15}=3}
- \latex{15\div \left( 15+30 \right)=\frac{15}{45}=\frac{1}{3}}
- What is the quotient of the durations of the morning and evening walks?
- How many times longer are the two walks combined than the morning walk with Ben?
- How many times longer is the evening walk than the morning walk?
- What fraction of the walks does Chips spend with Ben?
- By what number should the time spent with Zack be divided to get the time spent with Ben?
{{exercise_number}}. The image shows the first few terms of a sequence of polygons. Ask questions that can be answered based on the image.
{{exercise_number}}. The image shows the first few terms of a sequence of solids made up of small cubes. Ask questions that can be answered based on the image.
{{exercise_number}}. The images show various solids made up of small cubes. Come up with questions in connection with the solids, then answer them.
A)
B)
C)
Let’s play a guessing game.
One player thinks of an arbitrary number. It can be positive, negative, whole or a fraction.
The other player tries to guess the number by asking yes-or-no questions. The goal is to guess the number with as few questions as possible.
The other player tries to guess the number by asking yes-or-no questions. The goal is to guess the number with as few questions as possible.
Quiz
Ben showed Zack a picture and said, 'The son of this man is the son of my father". Who was in the picture?
