SOLVING WORD PROBLEMSUsing Algebraic Equations

TRANSLATIONS

plus / add subtract /minusgreater than difference /less

thanincrease / larger negative / decrease augment / sum reduce / diminish

times / of quotientmultiply divided by product a number of parts

TRANSLATIONS

equal / equal to / is / results in

gives you / makes / quotient / sumdifference / product

a number / an age / a quantity a distance / a width / a heighta mass / a volume / a price / a length

an amount / a number of coins

ENGLISH PHRASESA. 1) 18 + x eighteen plus a number 2) x + 18 a number plus eighteen 3) ¼ x one forth of a number 4) (13 +x) ÷ 3 thirteen plus a number the sum

is divided by 3

B 1) An unknown number squared 2) A number of coins tripled 3x 3) One-fifth of a number decreased by eight

4) Six times a number diminished by seven6x-7

CREATE ALGEBRAIC EQUATIONSC. 1) The square of a certain number diminished

by eight is greater than three times the number. 2) When a number of pennies is divided by fifty- two and thirty-nine is added to the quotient, the result is three hundred sixty-four.

3) Seven times a person’s age increased by nine is twenty-seven.

4) When fifteen kilometers are added to a certain distance, the result is thirty-six kilometers.

NEW LANGUAGEConsecutive Integers:

Consecutive Even Integers:

Consecutive Odd Integers:

Complementary Angles:

Supplementary Angles:

NUMBER PROBLEMS - NOTES 1. Identify what you are looking for and call it

‘x’.

2. Translate the problem into a mathematical statement (equation).

3. Solve the equation for ‘x’.

4. Use this knowledge of ‘x’ to answer the question. Use a final statement.

NUMBER PROBLEMS - EXAMPLE 1 If a number is tripled and the result increased

by four, the sum is sixty-four. Find the number.

Number = x

Equation 3x + 4 = 64

Solve 3x = 64 – 4 3x = 60 x = 60/3

x = 20Solution – the number is 20.

NUMBER PROBLEMS - EXAMPLE 2 The larger of two numbers is seven more

than five times the smaller number. If their sum is sixty-one, what are the numbers?

Larger Number = 5x +7 Smaller Number = xEquation 5x + 7 + x = 61Solve 6x +7 = 61

6x = 61 – 7 6x = 54

x = 54/6 x = 9

Solution – the small number is 9 and the larger is 5(9) + 7 = 52.

NUMBER PROBLEMS - EXAMPLE 3 The sum of three consecutive numbers is

seventy-eight. What are the numbers?

First : x Second: x + 1 Third: x +2Equation x + x + 1 + x + 2 = 78

Solve 3x + 3 = 78 3x = 78 – 3 3x = 75 x = 75/3 x = 25

Solution – The consecutive numbers are 25, 26, and 27.

NUMBER PROBLEMS - EXAMPLE 4 If three is added to a number and the sum is

multiplied by two, the result is the same as nine subtracted from three times the number. Find the number.

Number = xEquation 2(x + 3) = 3x – 9 Solve 2x + 6 = 3x – 9

2x – 3x = -9 – 6 - x = - 15 x = 15

Solution - The number is 15.

PRACTICE PROBLEMS Do questions 1- 12 on loose leaf.

You DO NOT need to copy the question, but follow the sequence to solve.

Do only questions 1-6 for modified students.

Check answers at the back.

GEOMETRIC PROBLEMS - NOTES 1. Identify what you are looking for and call it ‘x’ or

another variable.

2. Use a diagram and label parts that you know, including your variable that you don’t know.

3. Translate the problem into a mathematical statement (equation).

4. Solve the equation for your variable.

5. Answer the question. Use a final statement.Some equations are given on your sheet.

GEOMETRIC PROBLEMS - EXAMPLE 1 If the two equal sides of an isosceles triangle

are each five times as long as the base, and the perimeter is 253 metres, find the length of each side.

Base = x Side 1 = 5x Side 2 = 5xEquation x + 5x + 5x = 253Solve 11x = 253

x = 23

Solution: the base is 23m, and each of the sides are 5x23=115m.

5x 5x

x

GEOMETRIC PROBLEMS - EXAMPLE 2 Two angles of a triangle are congruent. The

third is twice as large as either of the other two What is the measure of each angle? (Remember – angles of a triangle add to 180°)

<1 = x <2 = x < 3= 2xEquation x + x + 2x = 180Solve 4x = 180

x = 45

Solution: two angles are 45°. And the other angle is 2(45) = 90°.

2x

xx

GEOMETRIC PROBLEMS - EXAMPLE 3 A farmer uses 54 hectometers of fencing to

enclose a rectangular field. If the width is two hectometers less than the width, find the dimensions of the field.

Length = x Width = x – 2 P = 54hmEquation 2(x + x – 2) = 54Solve 2(2x – 2) = 54

4x – 4 = 544x = 54 + 44x = 58

x = 14.5

Solution: the length is 14.5 hm and the width is 14.5 – 2= 12.5hm.

x - 2

x

AGE PROBLEMS - NOTES 1. Identify the unknown and call it ‘x’ or another

variable.

2. Use a chart to organize your information – including ages now, in the past and in the future.

3. Translate the problem into a mathematical statement (equation).

4. Solve the equation for your variable.

5. Answer the question. Use a final statement.

AGE PROBLEMS - EXAMPLE 1 A father is now only three times as old as his

son. Either years ago the father was five times as old as his son. Find their present ages.

Equation3x – 8 = 5(x-8) Solve 3x – 8 = 5(x-8)

3x – 8 = 5x – 403x – 5x = -40 + 8-2x = -32

x = 16Solution:.The son is 16 and the father is 3(16) = 48.

Father SonNow 3x xPast 3x – 8 x – 8

AGE PROBLEMS - EXAMPLE 2 John is twice as old as Bill. In five years, the

sum of their ages will be thirty-four.

Equation2x + 5 + x + 5 = 34 Solve 3x + 10 = 34

3x = 34 – 103x = 24

x = 8Solution:. Bill is 8 years old now, and John is 2(8) = 16 years old.

John BillNow 2x x

Future 2x + 5 x + 5

AGE PROBLEMS - EXAMPLE 3 A mother is three years less than five times

as old as her daughter. If the sum of their ages is thirty-nine, find the age of each.

Equation5x – 3 + x = 39 Solve 6x – 3 = 39

6x = 39 + 36x = 42

x = 7Solution:. The daughter is 7 years old and the mother is 5(7)-3 = 32 years old.

Mother DaughterNow 5x – 3 x