Solving worded problems

To solve worded problems:

express each unknown in terms of x

form an equation and solve it

give the answer in the context of the question.

EG+S

x 4

---x 4

---EG+S

In Q1 to Q5, form an equation and solve it to find the number.

1 a Six more than 4 times a number is equal to 18.

b Five less than twice a number is equal to 9.

c When a number is multiplied by 3, then increased by 7, the result is 43.

d Double a number, then reduce it by 9. The result is 13.

e When a number is multiplied by 5 and this is then subtracted from 60, the result is 20.

2 a A number is increased by 4 and then multiplied by 6. The result is 30.

b When 3 is subtracted from a number and this is then multiplied by 8, the result is 64.

c The difference between a number and 9 is tripled. The result is 45.

3 a If 12 is added to half of a number, the result is 25.

b A number is divided by 7, then decreased by 3. The result is 4.

c Two-thirds of a number is 18.

d Eight less than three-quarters of a number is 31.

4 a A number is added to 17 and then divided by 4. The result is 7.

b Think of a number. Double it, add 5, then divide by 3. The result is 11.

c A certain number is decreased by 4, doubled, then divided by 5. The result is 6.

5 a A number is doubled, then decreased by 9. The result is equal to 13 more than the number.

b Eighteen less than the product of a number and 5 is equal to double the number.

c Think of a number. Double it, add 20, then divide by 4. The result is equal to 3 times the number.

d One-third of a number is equal to 5 less than twice the number.

Example 3

The cost of a cricket ball is 80c more than the cost of a tennis ball. If 3 cricket balls and 4 tennis balls cost

$19.90, find the cost of each ball.

Solution

Let the cost of a tennis ball be x cents

∴ the cost of a cricket ball is (x + 80) cents.

3(x+ 80) + 4x = 1990 (NOTE: $19.90 = 1990c) 3x+ 240 + 4x = 1990

7x+ 240 = 1990

−240 −240 7x = 1750

÷7 ÷7

∴ x = 250

∴ Each tennis ball costs $2.50 and each cricket ball costs

$3.30.

EG+S

Exercise 4.7

■ Consolidation

6 Form an equation and solve it to answer each of the following.

a The sum of two consecutive numbers is 151. What are the numbers?

b The sum of three consecutive numbers is 54. What are the numbers?

c The sum of four consecutive numbers is 98. What are the numbers?

7 Form an equation and solve it to find the numbers in each of these.

a The sum of three consecutive even numbers is 102. Find the numbers.

b The sum of four consecutive odd numbers is 48. Find the numbers.

c The sum of two consecutive even numbers is equal to 27 more than the odd number that lies between them. Find the even numbers.

d The sum of three consecutive odd numbers is equal to 39 more than the sum of the even numbers that lie between them. Find the odd numbers.

8 Form an equation and solve it to find the value of the pronumeral in each of these.

9 Form an equation, then solve it to answer each of the following problems.

a In a group of 29 men and women, there are 7 more women than men. How many people of each gender are there?

b Annika has $9 less than Kris. If together they have $41, find the amount of money that each girl has.

c The perimeter of a parallelogram is 56 cm and one side is 6 cm shorter than an adjacent side. Find the lengths of the sides.

d The cost of a new tyre is $35 more than the cost of a retread. If the cost of two new tyres and two retreads is $370, find the unit cost of each tyre.

e An isosceles trapezium has two equal sides of length 7 cm. One of the parallel sides is 5 cm longer than the other parallel side. Find the lengths of the parallel sides if the trapezium has a perimeter of 35 cm.

f A 2.5 m length of timber is cut into 3 pieces. One piece is twice the length of the shortest piece and the other is 30 cm longer than the shortest piece. Find, in centimetres, the length of each piece of timber.

g Raymond is half the age of his father. The sum of their ages is 78 years. How old is each person?

10 Form an equation, then solve it to answer each of these.

a An imported brand of sugar costs 60c more per kilogram than an Australian brand. If 2 kg of imported sugar plus 5 kg of Australian sugar costs $13.80, find the cost per kilogram of the imported sugar.

(x – 5) cm

Perimeter = 36 cm

(2x + 7) cm

(x + 11) cm Perimeter = 85 cm

(5x + 3) cm 2x cm

Perimeter = 104 cm

a b c

b Jonathan is twice as old as Darren and Darren is three times as old as Bettina. The sum of their ages is 120 years. Find the age of each person.

c At a local fruit shop, tomatoes are sold at 24c each and pears are sold at 28c each. Keryn bought 8 more pears than tomatoes and paid the fruiterer $3.80. How many pears and tomatoes did Keryn purchase?

d If the numerator and denominator in the fraction are increased by a certain number, n, the value of the fraction would then be . Find the number.

e Penny has saved $18 in 20c and 50c coins. There are 8 more 50c coins than 20c coins.

What is the total value of the 20c coins?

f An apprentice mechanic agrees to be paid $90 for each day that he comes to work and to pay his employer $40 for each day that he does not come to work. How many days did the apprentice work in April if his total pay for the month was $1790?

■ Further applications

11 a A woman has a daughter who is half her age and a son who is two-thirds her age.

The sum of the children’s ages is 12 years more than the age of their mother. How old is each person?

b A man is 37 years old and his daughter is 5 years old. In how many years time will the man be 3 times the age of his daughter?

c Anita is 4 times as old as Frank. In 5 years time Anita will only be 3 times as old as Frank. Find their present ages.

d Six years ago, Wendy was twice the age of Thao. At present, Wendy is 30 years older than Thao. Find the present age of each woman.

12 Emma tries to guess the number of beads in a jar but guesses 75 too many. Laura guesses 63 too few. If the average of their guesses is 350, how many beads are in the jar?

13 A Boeing 729 airliner has a total mass at take-off 94 000 kg. The fuel and crew are the mass of the unloaded plane and the passengers and luggage are the mass of the fuel and crew. What is the mass of the unloaded plane?

5 11 ---2 3

---1 4 ---1

3