Mathematics Form 1-4 : CHAPTER FOURTY NINE – FORMULAE AND VARIATION

Specific Objectives

By the end of the topic the learner should be able to:

a) Rewrite a given formula by changing its subject

b) Define direct, inverse, partial and joint variations

c) Determine constants of proportionality

d) Form and solve equations involving variations

e) Draw graphs to illustrate direct and inverse proportions

f) Use variations to solve real life problems

Content

1. Change of the subject of a formula
2. Direct, inverse, partial and joint variation
3. Constants of proportionality
4. Equations involving variations
5. Graphs of direct and inverse proportion
6. Formation of equations on variations based on real life situations

Formulae

A Formula is an expression or equation that expresses the relationship between certain quantities.

For Example is the formula to find the area of a circle of radius r units.

From this formula, we can know the relationship between the radius and the area of a circle. The area of a circle varies directly as the square of its radius. Here is the constant of variation.

Changing the subject of a formulae

Terminology

In the formula

C =d

 Subject: C Rule: multiplyby diameter

The variable on the left, is known as the subject: What you are trying to find.

The formula on the right, is the rule, that tells you how to calculate the subject.

So, if you want to have a formula or rule that lets you calculate d, you need

to make d, the subject of the formula.

This is changing the subject of the formula from C to d.

So clearly in the case above where

C =d

We get C by multiplying by the diameter

To calculate d, we need to divide the Circumference C by

So d and now we have d as the subject of the formula.

Method:

A formula is simply an equation, that you cannot solve, until you replace the letters with their

values (numbers). It is known as a literal equation.

To change the subject, apply the same rules as we have applied to normal equations.

1. Add the same variable to both sides.

2. Subtract the same variable from both sides.

3. Multiply both sides by the same variable.

4. Divide both sides by the same variable.

5. Square both sides

6. Square root both sides.

Examples:

Make the letter in brackets the subject of the formula

x + p = q [ x ]

(subtract p from both sides)

x = q – p

y − r = s [ y ]

(add r to both sides)

y = s + r

P = RS [ R ]

(divide both sides by S)

S =

= L [ A ]

(multiply both sides by B)

A = LB

2w+ 3 = y [ w ]

(subtract 3 from both sides)

2w = y −3

(divide both sides by 2)

W=

P = Q [ Q ]

(multiply both sides by 3− get rid of fraction)

3P = Q

T = k [ k ]

(multiply both sides by 5− get rid of fraction)

5T = 2k

(divide both sides by 2)

= k Note that: is the same as

A =r [ r ]

(divide both sides by p)

(square root both sides)

L =h −t [ h ]

(multiply both sides by 2)

2L = h −t

(add t to both sides)

2L + t = h

Example

Make d the subject of the formula G=

Solution

Squaring both sides

Multiply both sides by d-1

Expanding the L.H.S

Collecting the terms containing d on the L.H.S

Factorizing the L.H.S

Dividing both sides by

Variation

In a formula some elements which do not change (fixed) under any condition are called constants while the ones that change are called variables. There are different types of variations.

• Direct Variation, where both variables either increase or decrease together
• Inverse or Indirect Variation, where when one of the variables increases, the other one decreases
• Joint Variation, where more than two variables are related directly
• Combined Variation, which involves a combination of direct or joint variation, and indirect variation

Examples

• Direct: The number of money I make varies directly (or you can say varies proportionally) with how much I work.
• Direct: The length of the side a square varies directly with the perimeter of the square.
• Inverse:   The number of people I invite to my bowling party varies inversely with the number of games they might get to play (or you can say is proportional to the inverse of).
• Inverse:  The temperature in my house varies indirectly (same as inversely) with the amount of time the air conditioning is running.
• Inverse:   My school marks may vary inversely with the number of hours I watch TV.

Direct or Proportional Variation

When two variables are related directly, the ratio of their values is always the same.  So as one goes up, so does the other, and if one goes down, so does the other.  Think of linear direct variation as a “y = mx” line, where the ratio of y to x is the slope (m).  With direct variation, the y-intercept is always 0 (zero); this is how it’s defined.

Direct variation problems are typically written:

→  y= kx where k is the ratio of y to x (which is the same as the slope or rate).

Some problems will ask for that k value (which is called the constant of variation or constant of proportionality ); others will just give you 3 out of the 4 values for x and y and you can simply set up a ratio to find the other value. 

Remember the example of making ksh 1000 per week (y = 10x)?  This is an example of direct variation, since the ratio of how much you make to how many hours you work is always constant.

Direct Variation Word Problem:

The amount of money raised at a school fundraiser is directly proportional to the number of people who attend.  Last year, the amount of money raised for 100 attendees was $2500. How much money will be raised if 1000 people attend this year?

Solution:

Let’s do this problem using both the Formula Method and the Proportion Method:

Formula method Explanation

Proportional method Explanation

Direct Square Variation Word Problem

Again, a Direct Square Variation is when y is proportional to the square of x, or  . 

Example

If yvaries directly with the square ofx, and if y = 4 when x= 3, what is y when x= 2?

Solution:

Let’s do this with the formula method and the proportion method:

Formulae method notes

Proportional method Notes

Example

The length (l) cm of a wire varies directly as the temperature c.The length of the wire is 5 cm when the temperature is .Calculate the length of the wire when the temperature is c.

Solution

l

Therefore l =Kt

Substituting l =5 when T= .

5 =k x 65

K =

Therefore l =

When t = 69

L =

Direct variation graph

Inverse or Indirect Variation

Inverse or Indirect Variation is refers to relationships of two variables that go in the opposite direction.  Let’s supposed you are comparing how fast you are driving (average speed) to how fast you get to your work.The faster you drive the earlier you get to your work. So as the speed increases time reduces and vice versa .

So the formula for inverse or indirect variation is:

→ y =    or K =xy  where k is always the same number or constant.

(Note that you could also have an Indirect Square Variation or Inverse Square Variation, like we saw above for a Direct Variation.  This would be of the form→ y =    or k=    .)

Inverse Variation Word Problem:

So we might have a problem like this:

          The value of yvaries inversely with x, and y = 4 when x = 3.  Find xwhen y = 6.

The problem can also be written as follows:

           Let = 3,  = 4, and  = 6.  Let yvary inversely as x.  Find.

Solution:

We can solve this problem in one of two ways, as shown.  We do these methods when we are given any three of the four values for x and y.

Product Rule Method:

Inverse Variation Word Problem:

For the club, the number of tickets Moyo can buy is inversely proportional to the price of the tickets.  She can afford 15 tickets that cost $5 each.  How many tickets can she buy if each cost $3?

Solution:

Let’s use the product method:

.

Example

If 16 women working 7 hours day can paint a mural in 48 days, how many days will it take 14 women working 12 hours a day to paint the same mural?

Solution:

The three different values are inversely proportional;  for example, the more women you have, the less days it takes to paint the mural, and the more hours in a day the women paint, the less days they need to complete the mural:

Joint Variation and Combined Variation

Joint variation is just like direct variation, but involves more than one other variable.  All the variables are directly proportional, taken one at a time.  Let’s do a joint variation problem:

Supposed x varies jointly with y and the square root of z.  When x = –18 and y = 2, then z = 9.  Find y when x = 10 and z = 4.

Combined variation involves a combination of direct or joint variation, and indirect variation.  Since these equations are a little more complicated, you probably want to plug in all the variables, solve for k, and then solve back to get what’s missing.  Here is the type of problem you may get:

      (a) yvaries jointly as x and w and inversely as the square of z.  Find the equation of variation when y = 100, x = 2, w = 4, and z = 20.

      (b) Then solve for y when x = 1, w = 5, and z = 4.

Solution:

Example

The volume of wood in a tree (V) variesdirectly as the height (h) and inversely as the square of the girth (g).  If the volume of a tree is 144 cubic meters when the height is 20 meters and the girth is 1.5 meters, what is the height of a tree with a volume of 1000 and girth of 2 meters?

Solution:

Example

The average number of phone calls per day between two cities has found to be jointly proportional to the populations of the cities, and inversely proportional to the square of the distance between the two cities.  The population of Charlotte is about 1,500,000 and the population of Nashville is about 1,200,000, and the distance between the two cities is about 400 miles.  The average number of calls between the cities is about 200,000. 

(a) Find the k and write the equation of variation.

(b) The average number of daily phone calls between Charlotte and Indianapolis (which has a population of about 1,700,000) is about 134,000.  Find the distance between the two cities.

Solution:

It may be easier if you take it one step at a time:

Math’s Explanation

Example

A varies directly as B and inversely as the square root of C. Find the percentage change in A when B is decreased by 10 % and C increased by 21%.

Solution

A= K

A change in B and C causes a change in A

= 1.21C

Substituting

=

Percentage change in A =

  =

= – 18

Therefore A decreases 18

Partial variation

The general linear equation y =mx +c, where m and c are constants, connects two variables x and y.in such case we say that y is partly constant and partly varies as x.

Example

A variable y is partly constant and partly varies as if x = 2 when y=7 and x =4 when y =11, find the equation connecting y and x.

Solution

The required equation is y = kx + c where k and c are constants

Substituting x = 2 ,y =7 and x =4, y =11 in the equation gives

7 =2k +c …………………..(1)

11 = 4k +c …………………(2)

Subtracting equation 1 from equation 2

4 = 2 k

Therefore k =2

Substituting k =2 in the equation 1

C =7 – 4

C =3

Therefore the equation required is y=2x +3

End of topic  

Past KCSE Questions on the topic.

1.  The volume Vcm³ of an object is given by

 V = 2 π r³1 – 2

  3   sc²

Express in term of π r, s and V

2.  Make V the subject of the formula

 T = 1 m (u² – v²)

  2  

3.  Given that y =b – bx² make x the subject

cx² – a

4.  Given that log y = log (10ⁿ) make n the subject

5.  A quantity T is partly constant and partly varies as the square root of S.

1. Using constants a and b, write down an equation connecting T and S.
2. If S = 16, when T = 24 and S = 36 when T = 32, find the values of the constants a and b,

6.  A quantity P is partly constant and partly varies inversely as a quantity q, given that p = 10 when q = 1.5 and p = 20, when q = 1.25, find the value of p when q= 0.5

7.  Make y the subject of the formula p = xy  

x-y

8.  Make P the subject of the formula

P² = (P – q) (P-r)  

9.  The density of a solid spherical ball varies directly as its mass and inversely as the cube of its radius

 When the mass of the ball is 500g and the radius is 5 cm, its density is 2 g per cm³

 Calculate the radius of a solid spherical ball of mass 540 density of 10g per cm³

10.  Make s the subject of the formula

√P = r 1 – as²

11.  The quantities t, x and y are such that t varies directly as x and inversely as the square root of y. Find the percentage in t if x decreases by 4% when y increases by 44%

12.  Given that y is inversely proportional to xⁿ and k as the constant of proportionality;

(a)  (i)  Write down a formula connecting y, x, n and k

(ii)  If x = 2 when y = 12 and x = 4 when y = 3, write down two expressions for k in terms of n.

Hence, find the value of n and k.  

(b)  Using the value of n obtained in (a) (ii) above, find y when x = 5 ¹/₃

13.  The electrical resistance, R ohms of a wire of a given length is inversely proportional to the square of the diameter of the wire, d mm. If R = 2.0 ohms when d = 3mm. Find the vale R when d = 4 mm.

14.  The volume Vcm³ of a solid depends partly on r and partly on r where rcm is one of the dimensions of the solid.

When r = 1, the volume is 54.6 cm³ and when r = 2, the volume is 226.8 cm³

(a) Find an expression for V in terms of r

(b) Calculate the volume of the solid when r = 4

(c) Find the value of r for which the two parts of the volume are equal

15.  The mass of a certain metal rod varies jointly as its length and the square of its radius. A rod 40 cm long and radius 5 cm has a mass of 6 kg. Find the mass of a similar rod of length 25 cm and radius 8 cm.

16.  Make x the subject of the formula

 P =  xy

  z + x

17.  The charge c shillings per person for a certain service is partly fixed and partly inversely proportional to the total number N of people.

(a)  Write an expression for c in terms on N

(b)  When 100 people attended the charge is Kshs 8700 per person while for 35 people the charge is Kshs 10000 per person.

(c)  If a person had paid the full amount charge is refunded. A group of people paid but ten percent of organizer remained with Kshs 574000.

Find the number of people.

18.  Two variables A and B are such that A varies partly as B and partly as the square root of B given that A=30, when B=9 and A=16 when B=14, find A when B=36.

19.  Make p the subject of the formula

A =  -EP  

√P² + N