T Madas T Madas What do we mean

What do we mean when we say two quantities are in proportion? It means

Directly proportional quantities: They increase or decrease at the same rate More formally: Two

Two variables v and t are directly proportional. When t = 8, v =18.

In a chemistry experiment, the reaction time t is directly proportional to the mass

What do we mean when we say two quantities are inversely proportional ? It

The Civic Centre is to be painted, so they call a firm of decorators.

INVERSELY PROPORTIONAL QUANTITIES One increases at the same rate as the other one decreases.

A variable P is inversely proportional to a variable A. When A = 2,

A variable F is inversely proportional to a variable t. When t = 3,

Sometimes we may be asked to set and solve problems involving direct or inverse

A variable A is directly proportional to the square of another variable r. When

A variable y is directly proportional to the SQUARE ROOT of another variable x.

A variable W is directly proportional to a variable m and inversely proportional to

A variable F is directly proportional to a variable m and inversely proportional to

Cost of packets of pens 3 pens cost £ 2 What does the graph

3 pens cost £ 2 Cost of packets of pens Number of pens 3

when graphed the points of Directly Proportional Quantities: 1. 2. always form a straight

u 3. 4 5. 44 6. 8 8. 16 10. 88 12. 92 14.

1 decorator takes 24 days to finish a job What does the graph of

1 decorator takes 24 days to finish a job No of decorators Days 1

The graphed points of Inversely Proportional Quantities: 1. always lie on a curve like

When plotted, Inversely Proportional quantities, always show as Hyperbolas. 4. 5 4 3. 5

Suppose we have a formula which contains 2 or more variables. The data which

The variable for which the formula is solved for is: Directly proportional to variables

Now a harder, worded proportionality problem © T Madas

60 workers, working a 9 hour day produce 720 toys a day. a ates

Three variables u, v and w are related by a formula. The following table

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What do we mean when we say two quantities are in proportion? It means that if: one of them doubles, the other one also doubles. one of them trebles, the other one also trebles. one of them x 4, the other one also x 4. one of them halves, the other one also halves. one of them ÷ 4, the other one also ÷ 4. Can you give examples of directly proportional quantities from every day life? © T Madas

Directly proportional quantities: They increase or decrease at the same rate More formally: Two variables are directly proportional if the ratio between them remains constant. © T Madas

Two variables v and t are directly proportional. When t = 8, v =18. Write a formula which links v and t, in the form v = … v t Proportional © T Madas

Two variables v and t are directly proportional. When t = 8, v =18. Write a formula which links v and t, in the form v = … v t v = kt This will be the formula when we find the value of k Proportionality Constant © T Madas

Two variables v and t are directly proportional. When t = 8, v =18. Write a formula which links v and t, in the form v = … v t v = kt 9 So: v = 4 t 9 t or: v = 4 v = kt 18 = k x 8 8 k = 18 8 = 2. 25 = 9 4 or: v = 2. 25 t © T Madas

In a chemistry experiment, the reaction time t is directly proportional to the mass m of the compound present. When the mass is 3 grams the reaction time is 0. 2 seconds. 1. Write a formula which links t and m, in the form t = … 2. What is the reaction time when the mass is 8 grams? t m t = km This will be the formula when we find the value of k Proportionality Constant © T Madas

In a chemistry experiment, the reaction time t is directly proportional to the mass m of the compound present. When the mass is 3 grams the reaction time is 0. 2 seconds. 1. Write a formula which links t and m, in the form t = … 2. What is the reaction time when the mass is 8 grams? t t = km 0. 2 = k x 3 3 k = 0. 2 m t = km 1 So: t = 15 m m or: t = 15 k = 0. 2 3 2 1 ≈0. 067 = 30 = 15 or: t ≈ 0. 067 m m using: t = 15 t = 8 15 ≈ 0. 53 s © T Madas

What do we mean when we say two quantities are inversely proportional ? It means that if: one of them doubles, the other one halves. one of them x 3, the other one ÷ 3. one of them x 4, the other one ÷ 4. one of them ÷ 2, the other one x 2. one of them ÷ 10, the other one x 10. Can you give an example of inversely proportional quantities from every day life? © T Madas

The Civic Centre is to be painted, so they call a firm of decorators. If this firm provide: 1 decorator will take 60 days for the job 2 decorators will take 30 days for the job 3 decorators will take 20 days for the job 4 decorators will take 15 days for the job 5 decorators will take 12 days for the job 6 decorators will take 10 days for the job 10 decorators will take 6 days for the job 12 decorators will take 5 days for the job 15 decorators will take 4 days for the job 20 decorators will take 3 days for the job 30 decorators will take 2 days for the job 60 decorators will take 1 day for the job 120 decorators will take ½ day for the job 1 x 60 2 x 30 3 x 20 4 x 15 5 x 12 6 x 10 10 x 6 12 x 5 15 x 4 20 x 3 30 x 2 60 x 1 120 x ½ © T Madas

INVERSELY PROPORTIONAL QUANTITIES One increases at the same rate as the other one decreases. More formally: Two variables are inversely proportional if their product remains constant. © T Madas

A variable P is inversely proportional to a variable A. When A = 2, P = 36. 1. Write a formula which links P and A, in the form P = … 2. Find the value of P when A is 2. 5. P 1 A Inversely Proportional © T Madas

A variable P is inversely proportional to a variable A. When A = 2, P = 36. 1. Write a formula which links P and A, in the form P = … 2. Find the value of P when A is 2. 5. P 1 A 1 P =kx. A Proportionality Constant © T Madas

A variable P is inversely proportional to a variable A. When A = 2, P = 36. 1. Write a formula which links P and A, in the form P = … 2. Find the value of P when A is 2. 5. P 1 A 1 P =kx. A k P =A This will be the formula when we find the value of k © T Madas

A variable P is inversely proportional to a variable A. When A = 2, P = 36. 1. Write a formula which links P and A, in the form P = … 2. Find the value of P when A is 2. 5. k P =A k 1 P A 1 36 = P =kx. A 2 k = 72 k P =A So: P = 72 A using P = 72 A 72 P = 2. 5 = 144 288 = 28. 8 = 5 10 © T Madas

A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. F 1 t Inversely Proportional © T Madas

A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. 1 F t F =k x 1 t Proportionality Constant © T Madas

A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. F F =k k F = t 1 t x 1 t This will be the formula when we find the value of k © T Madas

A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. 1 F t F =k x k F = t So: F = 36 t using F = 1 t 48 = 36 t 48 t = 36 k F = t k 12 = 3 k = 36 36 3 t = 48 = 4 © T Madas

A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. Since we do not require a formula in this example we could also have worked as follows: The product of inversely proportional quantities remains constant F x t = constant 12 x 3 = 36 48 x t = 36 36 ÷ 48 = 0. 75 © T Madas

Sometimes we may be asked to set and solve problems involving direct or inverse proportion to the: • square of a variable • cube of a variable • square root of a variable or simply combine 3 variables with direct and inverse proportion in the same problem. © T Madas

A variable A is directly proportional to the square of another variable r. When r = 3, A = 36. Find the value of A, when r = 2. 5 A r 2 A = kr 2 So: A = 4 r 2 using: A = 4 r 2 A = kr 2 36 = k x 32 9 k = 36 k =4 A = 4 x 2. 52 5 2 A =4 x 2 25 A =4 x 4 A = 25 © T Madas

A variable y is directly proportional to the SQUARE ROOT of another variable x. When x = 25, y = 3. Find the value of x, when y = 1. 2 y x y = k x 3 So: y = 5 x 3 using: y = 5 x 3 1. 2 = 5 x 5 x 6 = 3 x 5 5 6= 3 x x 5 y = k x 3 = k x 25 5 k = 3 3 k = 5 = 0. 6 x =2 x =4 © T Madas

A variable W is directly proportional to a variable m and inversely proportional to another variable t. When m = 2 and t = 8, W = 15. Find the value of W when m = 6 and t = 4. W m t x 1 t W=kx m t W = km t 60 m So: W = t using: W = 60 m t W = 60 x 6 4 W = 360 4 W = 90 W = km t 15 = k x 2 8 15 = 2 k 8 2 k = 120 k = 60 © T Madas

A variable F is directly proportional to a variable m and inversely proportional to the square of another variable r. When m = 10 and r = 2, F = 15. Find the value of F when m = 24 and r = 3. F m r 2 x 1 r 2 F =kx m r 2 F = km r 2 6 m So: F = 2 r using: F = 6 m r 2 F = 6 x 224 3 F = 144 9 F = 16 F = km r 2 15 = k x 10 22 15 = 10 k 4 10 k = 60 k = 6 © T Madas

Cost of packets of pens 3 pens cost £ 2 What does the graph of two directly proportional quantities looks like? © T Madas

3 pens cost £ 2 Cost of packets of pens Number of pens 3 6 9 12 15 18 Cost (£) 2 4 6 8 10 12 Let us plot the information of this table in a graph © T Madas

3 pens cost £ 2 Cost of packets of pens Number of pens 3 6 9 12 15 18 Cost (£) 2 4 6 8 10 12 £ 12 10 8 6 4 2 0 4 8 12 16 20 24 pens © T Madas

when graphed the points of Directly Proportional Quantities: 1. 2. always form a straight line through the origin 3. the line is a diagonal of every rectangle opposite corner is at the origin. £ 12 10 8 6 4 2 0 4 8 12 16 20 24 pens © T Madas

u 3. 4 5. 44 6. 8 8. 16 10. 88 12. 92 14. 28 17 v 5 8 10 12 16 19 21 25 The data above has been obtained from a chemistry experiment and concerns two quantities, u and v. Are u and v directly proportional quantities? © T Madas

u 3. 4 5. 44 6. 8 8. 16 10. 88 12. 92 14. 28 17 v 5 8 10 12 16 19 21 25 v 20 15 10 5 0 the quantities u and v are directly proportional 5 10 15 20 25 u © T Madas

u 3. 4 5. 44 6. 8 8. 16 10. 88 12. 92 14. 28 17 v 5 8 10 12 16 19 21 25 u v What is the gradient of the line? gradient = 20 the ratio between directly proportional quantities remains constant. 15 Work the ratio v : u from the table and compare it with the gradient of this line. 25 10 What would have happened if we plotted the data with the axes the other way round? 5 0 25 diff in y = ≈ 1. 47 diff in x 17 5 17 10 15 20 25 u v © T Madas

u 3. 4 5. 44 6. 8 8. 16 10. 88 12. 92 14. 28 17 v 5 8 10 12 16 19 21 25 u What is the gradient of the line? gradient = 20 the ratio between directly proportional quantities remains constant. 15 10 17 5 0 17 25 diff in y = 1. 47 0. 68 = ≈ 25 diff in x 17 5 10 25 15 20 25 Work the ratio vu : : uv from the table and compare it with the gradient of this line. What would have happened if we plotted the data with the axes the other way round? v © T Madas

u 3. 4 5. 44 6. 8 8. 16 10. 88 12. 92 14. 28 17 v 5 8 10 12 16 19 21 25 We could obtain a formula linking u and v u = kv So: u = 0. 68 v u = kv 3. 4 = k x 5 5 k = 3. 4 5 = 0. 68 = 6. 8 10 The proportionality constant is the gradient of the line in the graph 1 u 0. 68 v ≈ 1. 47 u or v = © T Madas

1 decorator takes 24 days to finish a job What does the graph of two inversely proportional quantities looks like? © T Madas

1 decorator takes 24 days to finish a job No of decorators Days 1 2 24 12 3 4 6 8 8 6 4 3 12 24 2 1 20 days 15 10 15 decorators 20 25 © T Madas

The graphed points of Inversely Proportional Quantities: 1. always lie on a curve like the one shown below. 2. whose opposite corner is at the origin. 20 15 days a l o b r e 10 p y H 5 0 5 10 15 decorators 20 25 © T Madas

P A 5 6 8 9 12 15 20 24 18 15 11. 25 10 7. 5 6 4. 5 3. 75 The data above has been obtained from the physics department and concerns two quantities, P and A. Are P and A inversely proportional quantities? © T Madas

When plotted, Inversely Proportional quantities, always show as Hyperbolas. 4. 5 4 3. 5 3 2. 5 2 1. 5 1 0. 5 1 2 3 4 5 6 7 © T Madas

Suppose we have a formula which contains 2 or more variables. The data which produced this formula is not available. Is it possible to establish if variables are directly proportional or inversely proportional? This is how this is done. © T Madas

The variable for which the formula is solved for is: Directly proportional to variables which appear in the numerator of the R. H. S v =s t c s = vt v v s Inversely proportional to variables which appear in the denominator of the R. H. S s v and s are directly proportional 1 v and t are inversely proportional t s and t are directly proportional t © T Madas

The variable for which the formula is solved for is: Directly proportional to variables which appear in the numerator of the R. H. S V = mgh c V m= gh c h = V mg V V V Inversely proportional to variables which appear in the denominator of the R. H. S m V and m are directly proportional g V and g are directly proportional h V and h are directly proportional m 1 m and g are inversely proportional m 1 m and h are inversely proportional 1 h and g are inversely proportional h g © T Madas

The variable for which the formula is solved for is: Directly proportional to variables which appear in the numerator of the R. H. S Inversely proportional to variables which appear in the denominator of the R. H. S F = GMm r 2 F F F G M m F 1 r 2 F and G are directly proportional F and M are directly proportional F and m are directly proportional F is inversely proportional to the square of r To get relationships between any other 2 variables we appropriately rearrange the formula. © T Madas

The variable for which the formula is solved for is: Directly proportional to variables which appear in the numerator of the R. H. S V=4 3 πr 3 Inversely proportional to variables which appear in the denominator of the R. H. S Rearranging the formula for r gives: r = 3 V 4π 3 V r 3 V is directly proportional to the cube of r V π ariable Because π is not a v_______; onstant n______ umber π is a c_______ CHALLENGE V oot of __ r is directly proportional to the c___ ube r___ © T Madas

The variable for which the formula is solved for is: Directly proportional to variables which appear in the numerator of the R. H. S S =u+v t Inversely proportional to variables which appear in the denominator of the R. H. S S 1 S S u v S u + v S is directly proportional to the sum of u and v t S and t are inversely proportional Because u and v are not in a product © T Madas

60 workers, working a 9 hour day produce 720 toys a day. a ates rs ula ber ich e. Find 1. any rs How 2. number of hours they work h and the number of toys T they produce. are to produce 1020 toys? The formula must contain the 3 variables w, h and T Suppose that: the workers work a constant number of hours per day Then: the toys produced will also double If we double the workers, ______________ Toys and workers are directly proportional quantities T w Suppose that: we keep the number of workers constant Then: toys produced will also double doubling the hours they work, the ______________ Toys and hours are directly proportional quantities T h © T Madas

Three variables u, v and w are related by a formula. The following table gives some of the values that these three variables can take: u v w 4 8 12 16 120 120 2 4 6 8 30 20 15 12 1 1 2 3 4 5 Obtain the formula linking these variables, solved for u. u v © T Madas