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The steps of solving problems
The following steps should always be taken when you solve a math problem:
- Understand the problem (read the text of the exercise more than once, imagine it or act it out if necessary).
- What is the question?
- Examine the data (which is irrelevant, which is important, etc.).
- Look for relationships between the data and deduce the solution from them:
- draw a diagram;
- work backwards;
- use the balance method, etc.
- Check the solution.
- Read the question(s) again and answer them.
Example 1
Zoe and Dora spent the day collecting paper. By the end of the day, Zoe had collected \latex{ 53\; kg } more paper than Dora, who collected \latex{ 74 \;kg }. They walked \latex{ 5\; km } while collecting paper, and then they had to walk \latex{ 1.5 \;km } more to get home. On the way home, Zoe gave Dora \latex{ 28 \;kg } of her collected paper. After this, who had more paper, and by how much?
Solution 1
Question:
Who had more paper, and by how much?
Data:
Data:
The original amount of paper
After Dora received 28 kg of paper from Zoe
Zoe
Dora
\latex{ 53 \,kg } more than Dora
\latex{ 28 \,kg } less than the original amount
\latex{ 74 \,kg }
\latex{ 28 \,kg } more than the original amount
The information about how much they walked while collecting paper and on their way home is irrelevant regarding the question of the exercise.
Relationships between the data:
Zoe collected \latex{ 53 \;kg } more paper than Dora.
Therefore, Zoe collected \latex{ 74 + 53 = 127\; kg } of paper originally.
Therefore, Zoe collected \latex{ 74 + 53 = 127\; kg } of paper originally.
After giving Dora \latex{ 28 \;kg } of her collected paper, Zoe was left with \latex{ 127 - 28 = 99 \;kg } of paper, and Dora had \latex{ 74 + 28 = 102 \;kg } of paper.
Check:
Based on the following table.
The original amount of paper
After Dora received 28 kg of paper from Zoe
Zoe
Dora
\latex{ 127 \,kg }
\latex{ 99 \,kg = 127- 28 \,kg }
\latex{ 74 \,kg }
\latex{ 102\, kg = 74 +28\, kg }
Difference
\latex{ 53\, kg }
\latex{ 3 \,kg }
Answer:
Dora had \latex{ 3 \,kg } more paper than Zoe.
Solution 2
Draw a diagram.
The amount of
paper Dora had:
paper Dora had:
The amount of
paper Zoe had:
\latex{ 53 \,kg }
\latex{ 25 \,kg }
\latex{ 28\, kg }
After Dora received \latex{ 28 } \latex{ kg } of paper from Zoe:
The amount of
paper Dora had:
paper Dora had:
The amount of
paper Zoe had:
\latex{ 28\, kg }
\latex{ 3\;kg }
\latex{ 25 \,kg }
Dora ended up with \latex{ 3 \;kg } more paper than Zoe after receiving \latex{ 28 \;kg } of paper from her.
Check:
Solving a problem in more than one way can be considered as checking the answer. The result is the same as in Solution 1.
Answer:
Dora had \latex{ 3 \;kg } more paper than Zoe.
Example 2
A fisherman was asked how many fish he had caught. He answered, 'I was hoping to catch twenty fish; however, if I had caught three times as many fish as I actually did, I still would have caught two fewer fish than I was hoping for. ' How many fish did the fisherman catch?
Solution 1
Question:
How many fish did the fisherman catch?
Data:
Data:
The number of fish he was hoping to catch.
If you multiply the number of fish he actually caught by three and add two to the product, you get the number of fish he was hoping to catch.
Draw a diagram to illustrate the relationship between the data.
If you multiply the number of fish he actually caught by three and add two to the product, you get the number of fish he was hoping to catch.
Draw a diagram to illustrate the relationship between the data.
One line segment represents the number of fish the fisherman actually caught.
The number of fish
the fisherman caught:
the fisherman caught:
The number of fish the
fisherman was hoping to catch:
\latex{ 2 }
\latex{ 20 }
The diagram shows that the number of fish the fisherman caught is
\latex{(20 -2)\div3 = 18\div3 = 6.}
Check:
The fisherman caught six fish. If he had caught three times as many fish, he would have caught \latex{ 18 }, which is two fewer than \latex{ 20 }, the number of fish he was hoping to catch.
Answer:
The fisherman caught six fish.
Solution 2
Work backwards.
The fisherman was hoping to catch \latex{ 20 } fish, which is two more than three times the number of fish he actually caught. Thus, three times the number of fish he caught is \latex{ 18 }, so the number of fish he actually caught is \latex{ 18 ÷ 3 = 6 }.
\latex{\times3}
\latex{\div3}
\latex{+2}
\latex{-2}
\latex{20}
\latex{6}
caught
hoped
\latex{18}
Check:
The result is the same as in Solution 1.
Answer:
The fisherman caught six fish.
Example 3
Replace the letters in the circles with numbers so that they correspond to the operations on the arrows.
\latex{ A }
\latex{ B }
\latex{ C }
\latex{-2}
\latex{\times3}
\latex{\div2}
Solution
Question:
What numbers can replace the letters in the circles?
Data:
The image shows that
- \latex{ B } is half of \latex{ A }; thus, \latex{ A } is equal to \latex{ B } multiplied by two;
- \latex{ C } is equal to \latex{ B } multiplied by three;
- \latex{ A } is two less than \latex{ C }; thus, \latex{ C } is two more than \latex{ A }.
Make a drawing.
The relationship between the data can be represented by line segments.
The line segments show that if \latex{ B } multiplied by three is two more than \latex{ B } multiplied by two, then the value of \latex{ B } can only be two.
Therefore \latex{A= 2 \times2 = 4} and \latex{C= 3 \times2 = 6.}
\latex{ B: }
\latex{ A: }
\latex{ C: }
\latex{ 2 }
Check:
The resulting numbers correspond to the operations on the arrows.
Answer:
Thus \latex{ A= 4}, \latex{ B= 2 } and \latex{ C= 6 } .
\latex{ 4 }
\latex{ 2 }
\latex{ 6 }
\latex{-2}
\latex{\times3}
\latex{\div2}
Exercises
{{exercise_number}}. When Tom was born, his mom was seven \latex{ years } younger than his dad. Tom is now twelve \latex{ years } old, and his father is \latex{ 44 } \latex{ years } old. What is the age difference between his parents now?
{{exercise_number}}. Replace the letters in the circles with numbers so that they correspond to the operations on the arrows.
a)
b)
\latex{ A }
\latex{ A }
\latex{ B }
\latex{ B }
\latex{ C }
\latex{-24}
\latex{+165}
\latex{\times3}
\latex{\times5}
\latex{\div4}
{{exercise_number}}. The bar graph shows the results of a math test taken by the sixth graders. Each student passed the exam. Based on the bar graph, determine how many students received \latex{ A }, \latex{ B }, \latex{ C }, \latex{ D } and \latex{ E } on the test.
Some students who received a \latex{ B } were only one point away from getting an \latex{ A }. If those students had received one more point, the number of students who got an \latex{ A } would have been equal to the number of students who got a \latex{ B }.
How many students needed one more point to get an \latex{ A } on this test?
\latex{ 10 }
\latex{ 5 }
\latex{ 0 }
number of students
\latex{ A }
\latex{B }
\latex{ C }
\latex{ D }
\latex{ E }
grade
{{exercise_number}}. Judith and Matt received the same number of votes in the election for student body president. Consequently, a new election was held in which some of the students changed their votes. Thirteen students voted for Judith instead of Matt, and \latex{ 37 } students voted for Matt instead of Judith. Who became the student body president? How many more votes did they receive than their opponent?
{{exercise_number}}. Pete and Paul are brothers. The sum of their ages is eleven. Pete is ten \latex{ years } younger than Paul. How old are the boys?
{{exercise_number}}. What is the length of the longer side of a rectangular garden whose shorter side is one-third of the longer side and requires a \latex{ 288\; m } long fence to surround it?
{{exercise_number}}. Dan spent his holiday by the sea. For one-third of the \latex{ days }, it rained, so he could not go swimming. For half of the remaining \latex{ days }, the weather was windy, so Dan could not swim in the sea due to the huge waves. There were only four \latex{ days } in which Dan was able to swim in the sea. How many \latex{ days } long was Dan's vacation?
{{exercise_number}}. There are three kids of unknown ages. The age of the oldest kid is twice the sum of the ages of the other two kids, and the middle kid is twice as old as the youngest. The sum of the ages of the three children is \latex{ 18 }. What are the ages of the kids?
{{exercise_number}}. The sum of the ages of the members in a family of five is \latex{ 123 }, and every member's age is a whole number. The dad is three \latex{ years } older than the mom, and each child is two \latex{ years } younger than the one before them. The dad's age multiplied by two is ten less than \latex{ 100 }. If there are no twins among the children, what was the sum of the ages of the family members four \latex{ years } ago?
Quiz
Divide \latex{ 45 } into four parts so that if you add two to the first part, subtract two from the second part, multiply the third part by two and divide the fourth part by two, all four parts become equal to each other.