INTEGRATED MATHEMATICS NUMBERS, GRAPHS & STATISTICS …

INTEGRATED MATHEMATICSNUMBERS, GRAPHS & STATISTICSFor

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2ALEX LIMMaster Trainer atMavis Tutorial Centre

02_Integrated Maths Numbers S2_TP.pdf 1 9/11/2016 10:41:19 AM

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ISBN-13 978-981-3210-44-8 ISBN-10 981-3210-44-3

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Integrated Mathematics NUMBERS, GRAPHS & STATISTICSfor Secondary 2

New Edition 2018

© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre

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Numbers, Graphs and Statistics

1 Direct and Inverse Proportion ...................................................................1

2 Graphs and Functions ..............................................................................13

3 Map Scales ...............................................................................................33

4 Dot Diagrams and Stem and Leaf Diagrams ...........................................47

5 Histograms ...............................................................................................75

6 Mean, Mode and Median .......................................................................109

7 Probability .............................................................................................135

solutions ................................................................................................ S1 - S52

01 Pre&Cont_Int Math Num,Graph&S3 3 9/22/2017 3:49:46 PM

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�Integrated Mathematics NuMbers, Graphs & statIstIcsfor secondary 2© singapore asia publishers pte Ltd & Mavis tutorial centre

Direct and Inverse Proportion1

Direct proportionWhen two quantities, A and B, are in direct proportion, • A = kB, where k is a constant and k ≠ 0,• the graph of A = kB is a straight line passing

through the origin.

two quantities, a and (b + 4)2 are in direct proportion and b is positive. some values of a and b are shown in the table below.

a x 1296 2025b 5 8 y

(a) Find the equation connecting a and (b + 4)2. (b) Hence, find the values of x and y.

Solutions:(a) since a and (b + 4)2 are in direct proportion, then a = k(b + 4)2,

where k is a constant. When a = 1296, b = 8, 1296 = k(8 + 4)2

1296 = k(144)

k = 1296 _____ 144 k = 9 thus, a = 9(b + 4)2.(b) When a = x, b = 5, x = 9(5 + 4)2

x = 729When a = 2025, b = y, 2025 = 9(y + 4)2

(y + 4)2 = 2025 _____ 9

(y + 4)2 = 225 y + 4 = 15 or y + 4 = –15 y = 11 y = –19 (rejected as y > 0)

Example 1

Topic 1

two quantities, A and B, are in direct proportion if A increases in a certain factor, and B also increases in that same factor. similarly, a certain decrease of A will lead to the same decrease in B.

the question states that y is positive, thus we reject −19.

02_T1_T3_Int Math Num,Graph&Stat1 1 9/22/2017 3:47:05 PM

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�Integrated Mathematics NuMbers, Graphs & statIstIcsfor secondary 2© singapore asia publishers pte Ltd & Mavis tutorial centre Topic 1

Given that y is directly proportional to the square of (x – 2), and that y = 12 when x = 4,(a) express y in terms of x,(b) find the value of y when x = 18,(c) find the possible values of x when y = 432.

Solutions:(a) since y is directly proportional to the square of (x – 2), then

y = k(x – 2)2, where k is a constant. When y = 12, x = 4, 12 = k(4 – 2)2

12 = k(4) k = 12 ___ 4 k = 3 thus, y = 3(x – 2)2.(b) When x = 18, y = 3(18 – 2)2

y = 768(c) When y = 432, 432 = 3(x – 2)2

(x – 2)2 = 432 ____ 3 (x – 2)2 = 144 x – 2 = 12 or x – 2 = −12 x = 14 x = −10

Example 2

It is given that y varies directly as x2 and y = 144 when x = n. Find the value of y when the value of x is doubled.

Solutions:since y varies directly as x2, then y = kx2, where k is a constant.When y = 144, x = n, 144 = kn2

k = 144 ____ n2 thus, y = 144 ____ n2 x2.

since the value of x is doubled, the new value of x will be 2n.When x = 2n, y = 144 ____ n2 (2n)2

y = 144 ____ n2 (4n2)

y = 576thus, the new value of y is 576.

Example 3


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� two quantities, x and y, as shown in the table below, are in direct proportion.

x 2 8 12 ny 4 16 m 96

(a) Write down an equation connecting x and y. (b) calculate the values of m and n.

� the length, l, of a spring varies directly as the amount of mass, m. a mass of 9 kg stretches the spring to a length of 9 cm. how long will the spring be if a mass of 7 kg is attached to it?

� Given that y is directly proportional to 3 √__

x and that y = 12 when x = 27, (a) find the equation connecting x and y. (b) Hence, find the value of x when y = 20.

Let’s Practise Direct proportion

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Inverse proportionWhen A and B are in inverse proportion,• A = k __ B , where k is a constant and k ≠ 0,• the graph of A against B is a hyperbola.

x 2 m 4y 12 8 n

the table above shows some values of x and the corresponding values of y.Given that x and y are in inverse proportion, (a) express y in terms of x,(b) find the values of m and n,(c) sketch a graph of y against x.

Solutions:

(a) since x and y are in inverse proportion, then y = k __ x , where k is a constant.

When x = 2 and y = 12,

12 = k __ 2 k = 2(12) k = 24

thus, y = 24 ___ x .

(b) When x = m, y = 8, (c) y

xo

y = 24 ___ x 8 = 24 ___ m 8m = 24 m = 24 ___ 8 m = 3

When x = 4, y = n,

n = 24 ___ 4 n = 6

Example 4

Topic 1

two quantities, A and B, are in inverse proportion if A increases in a certain factor, and B decreases in that same factor. similarly, a certain decrease of A will lead to the same increase in B.


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�Integrated Mathematics NuMbers, Graphs & statIstIcsfor secondary 2© singapore asia publishers pte Ltd & Mavis tutorial centre Topic 1

Given that y is inversely proportional to (x – 4) and y = −3 when x = 2. Find the value of y when x = 29.

Solutions:

since y is inversely proportional to (x – 4), then y = k _____ x – 4 , where k is a constant.

When y = −3 and x = 2, –3 = k _____ 2 – 4

–3 = k ___ –2

k = (–2)(–3) k = 6thus, y = 6 _____ x – 4 .

When x = 29, y = 6 ______ 29 – 4

y = 6 ___ 25

Example 5

Given that y is inversely proportional to x2 and y = 30 for a particular value of x.

calculate the value of y when x is doubled.

Solutions:

since y is inversely proportional to x2, then y = k __ x2 , where k is a constant.When y = 30 and x = n, 30 = k __ n2

k = 30n2

so y = 30n2 ____ x2 .

When x is doubled, x = 2n, y = 30n2 _____ (2n)2

y = 30n2 ____ 4n2

y = 7.5When x is doubled, y = 7.5.

Example 6


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� the table below shows the number of units of current, I, passing through a wire with resistance, r units. r is inversely proportional to I.

Number of units of current (I) 1 2 3Number of units of resistance (r) 30 15 10

(a) Write down the equation connecting I and r. (b) Find the current, I, passing through a wire with resistance 60 units. (c) Find the number of units of resistance a wire has when the current is 45 units.

� pressure (P) is inversely proportional to volume (V). Given that the pressure is 0.5 units when the volume is 64 cm3. Find

(a) the formula connecting P and V, (b) the volume when P = 72 units.

� Given that y is inversely proportional to the square root of (x – 3) and that y = 1 when x = 7. express y in terms of x.

Let’s Practise Inverse proportion

Topic 1

7 It is given that x is inversely proportional to the square root of y. Given that x = 3 when y = 25, find (a) the equation connecting x and y, (b) the value of x when y = 676, (c) the value of y when x = 9.

Let’s Practise Mixed practice


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7Integrated Mathematics NuMbers, Graphs & statIstIcsfor secondary 2© singapore asia publishers pte Ltd & Mavis tutorial centre

8 corresponding values of m and (n – 3)3 are given in the following table. m is inversely proportional to (n – 3)3.

m −3 0.375 x(n – 3)3 −1 y 125

(a) express m in terms of n. (b) Find the values of x and y.

9 Which one of the following figures represents a relationship of an inverse proportion? state a reason for the choice.

Figure 1O y2

x

Figure 2O y2

x

Figure 3O y2

x

�0 Given that r is inversely proportional to the cube root of s and that r = 1 1 __ 2 when s = 216, find(a) r in terms of s,(b) the value of s when r = 2.25.

�� It is given that y is inversely proportional to the cube of x. Given that y = 4 ___ 27 when x = 3, find

(a) the equation connecting x and y, (b) y when x = 8, (c) x when y = − 1 __ 2 .

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�� Given that A and l are in inverse proportion and A = 0.375 when l = 4. Find (a) the equation connecting A and l, (b) the value of A when l = 2, (c) the value of l when A = 3 ___ 32 .

�� study the table below carefully.x2 2 4 6 ay 0.5 1 1.5 5

(a) state if the above 2 quantities, x and y, are in direct or inverse proportion.(b) Write down the equation connecting x and y.(c) Write down the value of a.

�� Given that m is inversely proportional to the square root of n and m is 8 for a certain value of n, find the value of m when the value of n is 4 times its original value.

�� r is directly proportional to t3 and the difference in values of r when t = 7 and t = 5 is 1526.

(a) Form an equation connecting r and t3. (b) Find the value of t when r = −1512.

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�� m is directly proportional to n2 and m = 196 when n = 4. (a) Write down an equation relating m and n. (b) Find the values of n when m = 9216.

�7 y is directly proportional to the cube of x and y = 4096 when x = 8. Find (a) the equation relating y and x, (b) the value of y when x = 1 __ 4 , (c) the value of x when y = 64.

�8 the cost of a diamond, $C in thousands, is directly proportional to its size, n carats. If a diamond of 0.03 carat is $300, find

(a) an equation connecting C and n, (b) the size of a diamond, in carats, which costs $172 000.

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�0Integrated Mathematics NuMbers, Graphs & statIstIcsfor secondary 2© singapore asia publishers pte Ltd & Mavis tutorial centre

�9 It is given that x is directly proportional to y2 and x = 8820 when y = 14. (a) Find (i) an equation connecting x and y, (ii) the value of x when y = 17. (b) sketch the graph of the equation found in (a).

�0 A is directly proportional to c3 and A = 73 695 when c = 17. Find the value of c when A = – 643125.

�� y is directly proportional to x2. If x increases by 25%, find the percentage increase in y.

�� 16 men took 5 days to build an 8-km railway track. How long would it take 64 men to build a 128-km railway track?

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��Integrated Mathematics NuMbers, Graphs & statIstIcsfor secondary 2© singapore asia publishers pte Ltd & Mavis tutorial centre

�� 24 workers can build a road in 25 days. (a) how many days are required to build the same road if 4 workers are sick? (b) How many fewer workers are required to build the same road in 40 days?

�� 50 men take 38 days to finish building a bridge. Assuming that the men work at the same rate, how many more men must be hired to complete the bridge 13 days earlier?

�� 16 men can paint a building in 30 days. (a) how long would it take 6 men to paint the same building? (b) If the building needs to be painted in x days, how many men would be needed to

paint the building? express your answer in terms of x.

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��Integrated Mathematics NuMbers, Graphs & statIstIcsfor secondary 2© singapore asia publishers pte Ltd & Mavis tutorial centre

�� In an experiment, enough oxygen is pumped into a container to last for 10 hours for a group of 30 cockroaches.

(a) If the number of cockroaches is increased to 40, how many hours will the same amount of oxygen last?

(b) If 5 cockroaches are taken out of the container after 3 hours, how many hours and minutes can the oxygen last for the remaining 25 cockroaches?

�7 120 workers were employed to build a railway track 6 km long in 2 months. After 40 working days, only 3.6 km of the railway track was finished. How many more workers should be added in order to finish the remaining length of track on time? (Assume that a month consists of 30 days and all the workers worked at the same rate.)

�8 y is inversely proportional to x2. If the value of x decreases by 20%, find the percentage increase in y.

�9 It is given that the force between two particles is inversely proportional to the square of the distance between them. If the force is F when the distance between them is d, find the amount of the force, in terms of F, when the distance is increased by 215%.

�0 6 men were required to build a bridge 20 metres long in 18 days. On the fourth day, 2 men fell ill and thus could not work for 2 days. On the fifth day, they were tasked to extend the bridge for an additional of 5 metres in 10 days’ time. How many more men should be hired to complete the orders within the deadline?

Try These!

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S�Integrated Mathematics NuMbers, Graphs & statIstIcsfor secondary 2© singapore asia publishers pte Ltd & Mavis tutorial centre

Topic 1

�. (a) y = kx, where k is a constant. at (2, 4), 4 = k(2) k = 2 ∴ y = 2x (b) at (12, m), m = 2(12) m = 24 at (n, 96), 96 = 2n 2n = 96 n = 48

2. l = km, where k is a constant When l = 9 and m = 9, 9 = k(9) k = 1 ∴ l = m When m = 7, l = 7. ∴ the length of spring will be 7 cm.

3. (a) y = k 3 √__ x , where k is a constant.

at (27, 12), 12 = k 3 √___

27 12 = 3k k = 4 ∴ y = 4 3 √__

x (b) When y = 20, 20 = 4 3 √__

x 5 = 3 √__

x x = 53

x = 125

4. (a) r = k _ I , where k is a constant.

When I = 1, r = 30, 30 = k _ I k = 30 ∴ r = 30 ___ I

(b) When r = 60, 60 = 30 ___ I

I = 30 ___ 60

I = 1 __ 2 unit

∴ the current is 1 __ 2 unit.

(c) When I = 45, r = 30 ___ 45

r = 2 __ 3 unit

∴ the resistance is 2 __ 3 unit.

5. (a) P = k __ V , where k is a constant.

When P = 0.5, V = 64cm3, 0.5 = k ___ 64 k = 0.5(64) = 32 ∴ P = 32 ___ V

(b) When P = 72, 72 = 32 ___ V

V = 32 ___ 72

V = 4 __ 9 ∴ the volume is 4 __ 9 cm3.

6. y = k _____ √____

x – 3 , where k is a constant.

at (7, 1), 1 = k _____ √____

7 – 3

1 = k ___ √__

4 k = 2

∴ y = 2 _____ √____

x – 3

7. (a) x = k ___ √_ y , where k is a constant.

at (3, 25), 3 = k ____ √__

25

3 = k __ 5 k = 15

∴ x = 15 ___ √_ y

(b) When y = 676, x = 15 _____ √___

676

x = 15 ___ 26

(c) When x = 9, 9 = 15 ___ √_ y

√_ y = 15 ___ 9

y = ( 15 __ 9 ) 2 y = 2 7 __ 9

8. (a) m = k _______ (n – 3)3 , where k is a constant.

When m = –3, (n – 3)3 = –1, –3 = k ___ –1 k = (–3)(–1) k = 3 ∴ m = 3 _______ (n – 3)3

(b) When m = 0.375, (n – 3)3 = y, 0.375 = 3 __ y y = 3 _____ 0.375 y = 8

When m = x, (n – 3)3 = 125, x = 3 ____ 125

9. Figure 2 represents a relationship of an inverse proportion.

reason : a graph of hyperbola shape.

Topic 1

07 Solutions_Int Math Num,Graph&1 1 9/22/2017 3:22:18 PM

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INTEGRATED MATHEMATICS NUMBERS, GRAPHS & STATISTICS …