Problems
Often in mathematics (as well as in real life) we get a problem which is unlike any we have seen before. We need to reflect on what we already know and see how our existing knowledge can be used. Sometimes the problem will need us to develop new skills, or we may need to look at the problem in a different way.
Applying strategies is one of the processes involved in Working Mathematically.
Worked examples
Example 1
What is the angle between the hands of a clock at 2:25?
Solution 1
At 2 o’clock the angle between the hands is 60°.
In 60 minutes:
• the minute hand moves through 360°
• the hour hand moves through 30°.
In 25 minutes:
• the minute hand moves through
• the hour hand moves through
Hence, from the diagram:
60° + 12 ° + θ = 150°
∴ θ = 77 °
Maybe I can use the computer in some problems.
■ Some useful strategies for problem solving are:
• Eliminating possibilities
• Working backwards
• Acting it out
• Looking for patterns
• Solving a simpler problem
• Trial and error
• Making a drawing, diagram or model
• Using algebra
• Using technology
Start by drawing a diagram. Add
information to the diagram.
12 1
2 3 4 6 5
7 8 9
10 11
60 1212
25
60---×360° = 150°
25
60---×30° 121
2---°
=
1 2
---1 2
---Use problem-solving strategies to solve these problems.
a Vickie is expecting a baby. She must pick two given names (in order) for her child from the names she is considering. Girls’ names being considered are Rachel, Jessye, Faith and Kate.
Boys’ names are Jason, Brent and Grant. She is also considering the name Sandy for both a girl or a boy. How many ways of naming the child are being considered?
b If tyres sell for $96.70, $113.50, $125.90 and $143.30, which of the following amounts could be the cost of 5 tyres?
A $483.30 B $592.90 C $610 D $717.50
c Luke has 13 finches now but yesterday he lost of his finches when part of the roof blew off his aviary. The day before that he had given 6 zebra finches to his cousin and two days before that he had purchased two pairs of Gouldian finches. How many finches did Luke have before he bought the Gouldians?
Example 2
Screwdrivers come in four different sizes. The cost of these four sizes are $8.90, $7.80, $5.40 and $4.80.
I bought seven screwdrivers. Which of the following amounts was the total cost?
a $48.40 b $34 c $50.65 d $61.40
Solution 2
Try to eliminate possibilities.
• $50.65 cannot be the answer as the costs of all
screwdrivers are multiples of 10 cents. It’s impossible to get the 65 cents.
• Look at the maximum cost.
7 at $8.90 = $62.30
• $61.40 is less than $62.30 but close to it.
6 at $8.90 and 1 at $7.80 = $61.20
As all other combinations would cost less than $61.20, $61.40 cannot be the answer.
• Look at the minimum cost.
7 at $4.80 = $33.60
• $34 is not much more than $33.60.
6 at $4.80 and 1 at $5.40 = $34.20
As all other combinations would cost more than $34.20, $34 is not the answer.
• The only possibility remaining is $48.40, so it is the answer.
Note: We could continue to try different combinations of prices but we were told that one of the possibilities was correct and we have eliminated the other three.
Note: (2 × $8.90) + (2 × $7.80) + (1 × $5.40) + (2 × $4.80) = $48.40.
Exercise 2:02
1
3 4
---d One ‘move’ involves turning three coins over. What is the least number of ‘moves’ needed to change these five ‘tails’ to five ‘heads’?
e The difference of two numbers plus their sum is equal to 1. Write down one of these two numbers.
f 865 trees were planted in a row. Between the first and second trees 2 flowers were planted, between the second and third trees 1 flower, between the third and fourth 2 flowers, and so on to the end of the row. How many flowers were planted?
g We supply textbooks that have a mass six times as great as our summary books.
Five textbooks and ten summary books have a mass of 8 kg altogether. What is the mass of one textbook?
h What is the greatest number of points in which four circles can intersect?
For the chessboard on the right, how many squares are there of side length:
a 8 units? b 7 units?
c 6 units? d 5 units?
e 4 units? f 3 units?
g 2 units? h 1 unit?
Write the total number of squares as the sum of eight square numbers.
a Which two numbers have a sum of 79 and a product of 1288?
b Which two numbers have a difference of 11 and a product of 5226?
c Which two numbers have a sum of 1200 and a difference of 68?
a In our school’s House basketball competition each of the six teams must play the other five on two occasions. How many games must be played?
b A knockout tennis tournament is to be held for 53 players. How many byes must be given in the first round of the competition if the organisers do not want any byes in later rounds?
2
3
4
Plastic figures are needed for the doors on the 5th floor of a large hotel. If all of the numbers from 500 to 550 are needed, how many of each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 will be needed?
I have 20 Australian bank notes with a value of $720.
What could I have? (Give at least two solutions.)
A pentomino is formed by joining 5 squares together so that each square is joined to another square along an edge. How many different pentominoes are there? (Note: They are the same if one can be turned into the other by turning it upside down.)
Bill the gardener must buy some native plants. He can choose from Waratahs ($8), Grevilleas ($3) and Banksias ($5). He must choose at least 20 of each type and he must get at least 300 plants altogether. He also must not exceed his budget of $1200. Find a solution to his problem.
a Write the number 81 as the sum of:
i two consecutive integers ii three consecutive integers
b Find two other ways in which 81 can be written as the sum of a number of consecutive integers.
A number which is identical when its digits are written in reverse order is said to be palindromic; eg 21 412. The number 121 is a palindromic square number, since 121 = 112. a How many palindromic squares are there:
i with 1 digit (ie less than 10)?
ii with 2 digits (ie less than 100)?
iii with 3 digits (ie less than 1000)?
b Find how many palindromic square numbers there are that are smaller than 1 000 000?
(Hint: What could the last digit of a square number be?) 5
6
7
8
9
10
• How many times in a day are the hands of a clock at right angles?
Work out the answer to each part and put the letter for that part in the box that is above the correct answer.
H Find:
3 % of 1450 Simplify:
H −8 − (−16) I (− 4)2 L 64 ÷ (4 − 8) N − 1 − 1 − 1 O 1·25 × 100 O 30 × 0·2 P 7% of 300 R 125% of 10 S of (− 60)
T × 1
Special adhesive tape is used to mark the lines on the tennis court.
Use the dimensions given to calculate the length of tape required.
Assume the tape overlaps at each point of intersection.