Example 16
Solving a problem using equationsSolving a problem using equations
When Kate adds her current age and her age next year, the result is 19. How old is Kate now?
When Kate adds her current age and her age next year, the result is 19. How old is Kate now?
S SOOLLUUTTIIOONN EEXXPPLLAANNAATTIIOONN Let Letk k == K Kaattee’’s s ccuurrrreennt t aaggee.. DDeefifinne e a a pprroonnuummeerraal l tto o ssttaannd d ffoor r tthhe e uunnkknnoowwn n nnuummbbeerr.. k k ++ ( (k k ++ 1) 1)== 1 199 WWrriitte e aan n eeqquuaattiioon n tto o ddeessccrriibbe e tthhe e ssiittuuaattiioonn. . NNootte e tthhaatt k
k ++ 1 is Kate’s age next year. 1 is Kate’s age next year.
2 2k k ++ 1 1== 19 19 2 2k k == 18 18 k k == 9 9 − −11 ÷ ÷22 − −11 ÷ ÷22
Simplify the LHS and then solve the equation
Simplify the LHS and then solve the equation
systematically.
systematically.
EXTENSION
1
1 For each of the following problems, choose the best pronumeral definition.For each of the following problems, choose the best pronumeral definition.
a
a Problem: Monique’s age next year is 12. How old is she now?Problem: Monique’s age next year is 12. How old is she now?
A
A LetLetmm == Monique’s current age. Monique’s current age.
B
B LetLetmm == Monique. Monique.
C
C LetLetmm == 12. 12.
D
D LetLetmm == Monique’s age next year. Monique’s age next year.
E
E LetLetmm == this year. this year.
b
b Problem: Callan has 15 boxes, which weigh a total of 300 kg. How much does each box weigh?Problem: Callan has 15 boxes, which weigh a total of 300 kg. How much does each box weigh?
A
A LetLetww== 15. 15.
B
B LetLetww== 300. 300.
C
C LetLetww== the weight of one box. the weight of one box.
D
D LetLetww== the number of boxes. the number of boxes.
E
E LetLetww== the total weight. the total weight.
c
c Problem: Jared’s family has a farm with cows and sheep. The total number of animals is 200 andProblem: Jared’s family has a farm with cows and sheep. The total number of animals is 200 and
there are 71 cows. How many sheep are there?
there are 71 cows. How many sheep are there?
A
A LetLet x x == the size of a sheep. the size of a sheep.
B
B LetLet x x == the total number of the total number of animals.animals.
C
C LetLet x x == the number of sheep. the number of sheep.
D
D LetLet x x == the number of cows. the number of cows.
E
E LetLet x x == Jared’s age. Jared’s age.
2
2 Solve the following equations by inspection or systematically.Solve the following equations by inspection or systematically.
a
a 55 x x == 30 30 bb 77aa ++ 2 2== 16 16 cc 22k k −− 3 3== 15 15
Exercise 9H
Exercise 9H
E X T E N S I O NE X T E N S I O NW
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3 Launz buys a car and Launz buys a car and a trailer for a combined cost of a trailer for a combined cost of $40 $40 000. The trailer costs $2000.000. The trailer costs $2000.
a
a Define a pronumeral for the car’s cost.Define a pronumeral for the car’s cost.
b
b Write an equation to describe the problem.Write an equation to describe the problem.
c
c Solve the equation systematically.Solve the equation systematically.
d
d Hence, state the cost of the car.Hence, state the cost of the car.
4
4 Meghan buys 12 pens for a total cost of $15.60.Meghan buys 12 pens for a total cost of $15.60.
a
a Define a pronumeral for the cost of one pen.Define a pronumeral for the cost of one pen.
b
b Write an equation to describe the problem.Write an equation to describe the problem.
c
c Solve the equation systematically.Solve the equation systematically.
d
d Hence, state the cost of one pen.Hence, state the cost of one pen. Example 16
Example 16
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5
5 Jonas is paid $17 per hour and Jonas is paid $17 per hour and gets paid a bonus of $65 each week. One particular weekgets paid a bonus of $65 each week. One particular week
he earned $643.
he earned $643.
a
a Define a pronumeral for the number of Define a pronumeral for the number of hours Jonas worked.hours Jonas worked.
b
b Write an equation to describe the problem.Write an equation to describe the problem.
c
c Solve the equation systematically.Solve the equation systematically.
d
d How many hours did Jonas work in that week?How many hours did Jonas work in that week?
6
6 This rectangulaThis rectangular paddock has an area of 720 r paddock has an area of 720 mm22..
a
a Write an equation to describe the problem, usingWrite an equation to describe the problem, using �� for the for the
paddock’s length.
paddock’s length.
b
b Solve the equation systematically.Solve the equation systematically.
c
c How long is the paddock?How long is the paddock?
d
d What is the paddock’s perimeter?What is the paddock’s perimeter?
24 m 24 m
metresmetres
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7 A number is doubled, A number is doubled, then 3 is added then 3 is added and the result is doubled and the result is doubled again. This givagain. This gives a final resultes a final result
of 34. Set up and solve an equation to find the original number, showing all the steps clearly.
of 34. Set up and solve an equation to find the original number, showing all the steps clearly.
8
8 The perimeter of the shape shown is 30.The perimeter of the shape shown is 30.
Find the value of
Find the value of x x ..
9
9 Alexa watches some television on Monday, then twice as many hours on Tuesday, then twiceAlexa watches some television on Monday, then twice as many hours on Tuesday, then twice
as many hours again on Wednesday. If she watches a total of
as many hours again on Wednesday. If she watches a total of 1010 1 1
2
2 hours from Monday to hours from Monday to
Wednesday, how much television did Alexa watch on Monday?
Wednesday, how much television did Alexa watch on Monday?
2 2 x x x x 6 6 12 12 W
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10 Marcus and Sara’s combined age is 30. Given that Sara is 2 years older than Marcus, writeMarcus and Sara’s combined age is 30. Given that Sara is 2 years older than Marcus, write
an equation and find Marcus’ age.
an equation and find Marcus’ age.
11
11 An isosceles triangle is shown below. Write an equation and solve it to findAn isosceles triangle is shown below. Write an equation and solve it to find x x °°, the unknown angle., the unknown angle.
(Remember: The sum of angles in a triangle is
(Remember: The sum of angles in a triangle is 180180°°.).)
x x°° x x ° ° 154 154°° 12
12 Find the value ofFind the value of y y in the in the triangle shown here, by first writing an equation.triangle shown here, by first writing an equation.
y y°°
(2
(2 y y))°°
13
13 A rectangle has baseA rectangle has base bb and height and heighthh. The perimeter and area of the rectangle are equal. Write an. The perimeter and area of the rectangle are equal. Write an
equation and solve it by inspection to find
equation and solve it by inspection to find some possible values forsome possible values forbb and and hh. (Note: There are many. (Note: There are many
solutions to this equation. Try to find a few.)
solutions to this equation. Try to find a few.)
14
14 Find the values ofFind the values of x x and and y y in the rectangle shown. in the rectangle shown. 2 2 x x++33 3 3 y y 66 10 10 W W M M A A T T H H E E M
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15 If photocopying costs 35 cents a If photocopying costs 35 cents a page andpage and p p is the number of pages photocopied, which of the is the number of pages photocopied, which of the
following equations hav
following equations have possible solutions? Justify your e possible solutions? Justify your answers. (Note: Fraction answersanswers. (Note: Fraction answers
are not possible because you must still pay
are not possible because you must still pay 35 cents even if you photocopy only 35 cents even if you photocopy only part of a part of a page.)page.)
a
a 0.350.35 p p== 4.20 4.20 bb 0.350.35 p p== 2.90 2.90 cc 0.350.35 p p== 2.80 2.80
16
16 Assume that an isosceles triangle is drawn so that each of Assume that an isosceles triangle is drawn so that each of its three angles is a whole its three angles is a whole number ofnumber of
degrees. Prove that the angle
degrees. Prove that the angle aa must be an even number of degrees. must be an even number of degrees.
a a°° b b°° b b°° W
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