Chapter 18
Algebraic processes (5): Variation
Learning objectives
By the end of this chapter, the students should be able to:
1. Solve numerical and word problems involving direct and inverse variation.
2. Solve numerical and word problems involving joint and partial variation.
Teaching and learning materials
Students: Textbook, exercise book, writing materials.
Teacher: Wall charts with examples of the four
kinds of variation.
Teaching notes
• If x is the independent variable and y is the
dependent variable, they vary directly if there is
a constant number k ≠ 0 such that y = kx, where
k is the constant of variation. If two quantities
vary directly, we say that they have a direct
variation.
This means that if x is multiplied by a
number, then y is also multiplied by that same
number.
It also means that if the value of one quantity
increases, the value of the other quantity
increases by the same ratio.
• The variables x and y vary inversely for a
constant k ≠ 0, y = _kx or xy = k, where k is the
constant of variation and we say that the two
quantities have an inverse variation.
This also means that if a value of the
independent variable (x) is multiplied by 2, for
example, then the corresponding value of y is
multiplied by the multiplicative inverse of 2,
namely, _12 .
So, if the value of one of the variables is
multiplied by a number, the value of the other
variable is multiplied by the multiplicative
inverse of that number.
If two variables vary inversely, it means that
if the value of one increases, the value of
the other decreases in the same ratio or increases
inversely.
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• When we say c varies jointly to a set of
variables, it means that c varies directly and/
or inversely to each variable one at a time.
If c varies directly to a and inversely to b, the
equation will be of the form c = k_ba , where k is the
constant of variation and k ≠ 0.
The area, A, of a triangle, for example, varies
directly to the length of its base, b, and to the
length, h, of its perpendicular height.
The equation is A = kbh and k = _12 , therefore,
the final equation is A = _12 bh.
• When L varies partially to F, then L, is the sum
of a constant number and a constant multiple of
F. This is called partial variation.
The formula is then of the form L = kF + c,
where k and c are constants. We can also say that
k is the constant of variation and c is the initial
value of L.
We say L varies partially to F because in the
equation L = kF, L varies directly to F, but now
something is added.
If a force is applied to a spring, we know that
the increase in length varies directly to the force
applied to the spring.
If we want to get the length of the spring, we
have to add the increase in length of the spring
to its original length.
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Areas of difficulty and common mistakes
There are no specific areas of difficulty and
common mistakes in this work.
Chapter 18: Algebraic processes (5): Variation
9781292119748_ngm_mat_fm1_tg_eng_ng.indb 50
2015/08/02 2:06 PM