Workshop: Numeracy 3
Topics Covered:
• Algebraic operations
• Algebraic formulae
• Problem solving
by Dr.I.Namestnikova
[email protected]
Algebraic operations
Simplify the following expressions:
9x + 3 − 3x − 4 = = 9x − 3x + 3 − 4
= (9 − 3)x + (3 − 4)
= 6x − 1
5t − 3q − 7t − 2q + 1 = 5t − 7t − 3q − 2q + 1
= (5 − 7)t + (−3 − 2)q + 1
= −2t − 5q + 1
3t + 14 − 5t + 8 = 3t − 5t + 14 + 8 = −2t + 22
6yz − 4zy + y − 6z − 2y = 6yz − 4yz + y − 2y − 6z = 2yz − y − 6z
8mnt − 7tn + 5mt − 4mnt = 4mnt − 7tn + 5mt
5z − 2z + 7z + 2y + 3z − 7y = 13z − 5y
Algebraic operations
Simplify the following expressions:
5x × (−7) = −35x
5x × (−7)x = −35x2
4xz ÷ (2zx) =
4xz
2zx
3x × 3y × 2x ÷ y =
=
4xz
=2
2xz
3x3y2x
y
= (3x × 2x)
3y
y
= 18x2
10
9
10
x+4
10
÷
=
×
=
3x + 12
x+4
3x + 12
9
27
3(a + 2b) = 3a + 6b
4(2y − 5) − 3(2y + 8) = 8y − 20 − 6y − 24 = 2y − 44
(3y − 5) × (2y + 8) = 6y 2 − 10y + 24y − 40 = 6y 2 + 14y − 40
(7x + 1)(2y − 8) = 14xy − 56x + 2y − 8
Substitution
Replacing a variable with a number is called Substitution
• Find 142800 ÷ a, if a = 3, 100, 700
a=3
142800 ÷ a = 142800 ÷ 3
= 47600
a = 100 142800 ÷ a = 142800 ÷ 100 = 1428
a = 700 142800 ÷ a = 142800 ÷ 700 = 204
• If x = 3 find the value of 7x − 3
7x − 3 = 7 × 3 − 3 = 18
• If x = 1 and y = 5 find the value of 5x + 3xy
5x + 3xy = 5 × 1 + 3 × 1 × 5 = 20
• If x = −3, y = 3 and z = 2 find the value of 2(xz + 3y) − 4zy
2(xz + 3y) − 4zy = 2(−3 × 2 + 3 × 3) − 4 × 2 × 3 = −18
Exercises
N 16 kilometers is about 10 miles. About how many kilometers is 50 miles? (Answer: 80 km)
N 8 kilometers is about 5 miles. About how many kilometers is 5k miles? (Answer: 8k km)
N One carton of apple juice holds k cups of juice. How many cups can be filled from 30 cartons?
(Answer: 30k)
N A class bakes cakes. Emily bakes m cakes. Ben bakes 3 less cakes than Emily. Joe bakes 4 times
as many cakes as Emily. Loran bakes 2 more cakes then Joe. How many cakes each person bakes?
Hint: Use algebraic formulae.
Name
Cakes
Emily
m
Answer: Ben
m−3
Joe
4m
Loran
4m + 2
N Write a formula for the total time T taken to cook the Christmas turkey if the weight of the turkey is w
kg and you need to allow 20 minutes cooking time for each kilogram plus an extra 45 minutes.
(Answer:
T = 20w + 45)
Exercises
N The
numbers is 210. What is the third number if two of the numbers are m and n?
product of three
210
Answer:
mn
N My uncle was born in 1969. In what year will he be x yeas old? (Answer: 1969 + x)
N A coach holds n passengers.
How many
coaches will be needed for a school journey of tree hundred
340
and forty people? Answer:
n
N A test is forty questions, each worth one mark. The pass mark is 60%. How many questions have to
be answered correctly to pass the test?
60
Answer: 40
= 24
100
N A test is x questions, each worth y mark. The pass mark is 70%. How many questions have to be
answered correctly to pass the test?
Answer:
xy
70
100
= 0.7xy