REVISION (YR 9 ORANGE BOOK)
CHAPTER 1 – ALGEBRA
1 Write down the first four terms of each sequence whose nth term is given below.
a 3n + 1
b 4n – 2
c n2 + 7 d n(n + 3)
2 Find the nth term of each of the following sequences.
a 5, 7, 9, 11, …
b 2, 5, 8, 11, …
c 1, 4, 9, 16, …
d 3, 6, 11, 18, …
3 Find the nth term of each of the following sequences of fractions
a
1
2
,
2
3
,
3
4
, 54
b
1
3
,
2
5
,
3
7
,
4
9
4 Find the nth term of each of the following sequences.
a 3.5, 5, 6.5, 8,
b 5.1, 7.2, 9.3,
c 3.6, 6.1, 8.6, 11.1,
9.5,…
11.4, …
Look at the following diagrams.
a
b
c
d
e
…
Diagram 1 2 3 4 5 6
Crosses
1 5 13
Before drawing a diagram, can you predict, from the table, the number of crosses which are
in Diagram 4?
Draw Diagram 4, and count the number of crosses there are. Were you right?
Now predict the number of crosses for diagrams 5 and 6.
Check your results for part c by drawing diagrams 5 and 6.
Write down the term-to-term rule for the sequence of crosses. (Hint: 4 = 22, 8 = 23)
1 Write down the inverse of each of the following functions.
a x → 3x b x → x + 8
x → 3x – 5
c x→6+x
d x→
x
2
e x → 2x + 1
f x → 4x + 3
2 Write down two different types of inverse function and show that they are self-inverse
functions.
3 Write down the inverse of each of the following functions.
a x → 3(x + 5)
b x→
1
2
(x + 5)
c x →  6 4 x 
4 a On a pair of axes, draw the graph of the function x → 2x + 3.
b On the same pair of axes, draw the graph of the inverse of x → 2x + 3.
c Comment on the symmetries of the graphs.
g
1 Sketch graphs to show how the depth of water varies with time when water drips steadily into
the following containers.
a
b
c
2 Sketch distance-time graphs to illustrate each of the following situations.
a A car accelerating away from traffic lights. b A train slowing down to a standstill in a
railway station.
c A car travelling at a steady speed and then having to accelerate to overtake another
vehicle before slowing down to travel at the same steady speed again.
3 Sketch a graph to show the depth of water in a bath where it is filled initially with just hot
water, then the
old water is also turned on. After 2 minutes, a child gets into the bath, splashes about for 5
minutes
before getting out, and pulling out the plug. It takes 6 minutes for the water to drain away.
1 A sequence starting at 1 has the term-to-term rule Add 3 and divide by 2.
a Find the first 10 terms generated by this sequence.
b To what value does this sequence get closer and closer?
c Use the same term-to-term rule with different starting numbers. What do you notice?
2 Repeat Question 1, but change the term-to-term rule to Add 4 and divide by 2.
3 What would you expect the sequence to do if you used the term-to-term rule Add 7 and
divide by 2?
4 What will the sequence get closer to using the term-to-term rule Add A and divide by 2?
5 Investigate the term-to-term rule Add A and divide by 3.
CHAPTER 2 – NUMBER
Convert each of the following pairs of fractions to equivalent fractions with a common
denominator.
1 Convert each of the following pairs of fractions to equivalent fractions with a common
denominator. Then work out each answer, cancelling down and/or writing as a mixed number if
appropriate.
2
1
5
3
5
5
2
1
a 2 5  2 4 b 2 3  1 8 c 2 8  1 12 d 3 12  1 4
2 Work out each of the following. Cancel before multiplying when possible.
2 3
3
2
3
3
3
1
a 68
b 34
c 9  16
d 4 5 1 7
e 2 8 1 5
3 Work out each of the following. Cancel at the multiplication stage when possible.
1 1
3
9
5
1 1
7
3
3
a 43
b 16  14
c 63
d 2 8  16 e 2 5  10
1
3
1 How much would you have in the bank if you invest as follows?
a £450 at 3% interest per annum for 4 years.
b £6000 at 4.5% interest per annum for 7 years.
2 Stocks and shares can decrease in value as well as increase. How much would your stocks and shares be
worth if you had invested as follows?
a £1000, which lost 14% each year for 3 years.
b £750, which lost 5.2% each year for 5 years
1 A packet of biscuits claims to be 24% bigger! It now contains 26 biscuits. How many did it have
before the increase?
2 After a 10% price decrease, a hi-fi system now costs £288. How much was it before the decrease?
3 This table shows the cost of some items after 17 21 % VAT has been added. Work out the cost of each
item before VAT.
Item
Cost inc VAT
Item
Cost inc VATt
Radio
£112.80
Cooker
£329
Table
£131.60
Bed
£376
4 A pair of designer jeans is on sale at £96, which is 60% of its original price. What was the original
price?
5 A pair of boots, originally priced at £60, were reduced to £36 in a sale. What was the percentage
reduction in the price of the boots?
1 In 4 hours a man earns £45. How much does he earn in 5 hours?
2 A man walking one dog takes 20 minutes to walk one mile. How long will it take him to cover
onemile if he walks two dogs?
3 In a week, grass grows 21 mm. How much does it grow in 4 days?
4 Fifty litres of petrol costs £35. How much will 20 litres of petrol cost?
5 Eight men dig a ditch in 9 days. How long would six men take?
6 A camping party of three has enough food to last them 4 days. If another person joins the party, how
long will the food last?
7 At £6 an hour, Jack takes 16 hours to earn enough for a guitar. If he had earned £8 an hour, how
long would it have taken him to earn the money?
8 Three bell ringers ring a tune on 6 bells in 5 minutes. How long would four bell ringers take to ring
the same tune?
1 Two similar, plane shapes, A and B, have lengths in the ratio 1 : 4. The area of shape A is 10 cm2.
What is the area of shape B?
2 Two similar, plane shapes, P and Q, have lengths in the ratio 1 : 2. The area of shape Q is 100 cm2.
What is the area of shape P?
3 Two similar solids, C and D, have lengths in the ratio 1 : 3. The volume of solid C is 15 cm3.
What is the volume of solid D?
4 Two similar solids, R and S, have lengths in the ratio 1 : 2. The volume of solid S is 72 cm3.
What is the volume of solid R?
By rounding each value to one significant figure, estimate the answer to each of the following.
a 0.83 × 793
b 618 ÷ 0.32
c 812 ÷ 0.38
d 0.78 × 0.049
e (38 × 3.2) ÷ 0.487
f (2.7 + 6.3) × (0.5.2 – 0.17)
CHAPTER 3 – ALGEBRA
1 Solve the following equations.
a 5(m – 2) – 4(m + 3) = 0
b 5(k + 4) – 3(k – 6) = 0
c 4(y + 7) – 3(y + 5) = 0
d 3(2x – 4) – 2(4x + 5) = 0
2 Identify whether each of the following is an equation, a formula or an identity.
a 4x + 7 = 3x – 7 b (ab)2 = a2b2 c t = 8 + 9t
d W = 5q – R
e 12m = 4m + 8m f w + 7 = 3w – 1
3 The formula for the surface area of a cylinder is A = 2πrh + 2πr2.
Work out the surface area of these cylinders. Give your answer in terms of π.
a r = 10, h = 3
b r = r, h = 2r
4 Show that
x
y =
a x = 36, y = 4
x
y
is an identity by substituting into both sides:
b x = 4a2, y = a2
Solve simultaneously using elimination
1 4x + y = 14
2
2x + y = 8
3 3x + y = 10
4
8x – y = 1
5 5x – 4y = 36
6
2x – 4y = 6
6x + 3y = 33
2x + 3y = 21
5x + 2y = 22
7x – 2y = 2
5x + 3y = 50
9x – 3y = 48
Solve simultaneously using substitution
1 3x + y = 8
2x + 5y = 27
3 5x + 2y = 47
3x – y = 26
5 7x – 4y = 16
x–y=1
2 6x + 4y = 36
2x + y = 11
4 3x + y = 24
5x + 2y = 41
6 8x – 4y = 36
x + 3y = 8
1 Solve each of the following equations.
a
3x
 12 b
5
3t
6
5
c
6m
 18
8
d
2x
8
5
e
2w
6
7
2 Solve each of the following equations.
a
x 1
5
3
b
x5
8
4
2x  4
6 d
5
c
3x  1
2
8
3 Solve each of the following equations.
a
x 1 x 1
2x  3 x  2


b
3
4
3
2
c
3x  2 x  4

5
2
c
7
5

5 x  2 3x  5
4 Solve each of the following equations.
a
5
3

x 1 x 1
b
4
5

3x  2 2 x  1
1 Solve the following inequalities and illustrate their solutions on number lines.
a 5x + 7 ≥ 22
b 2x - 3 ≤ 10
d 2(x + 4) > 20
e 4(3t + 7) ≤ 16
c 4x + 3 < 11
f
2(5x – 4) ≥ 17
2 Write down the values of x that satisfy the conditions given.
a 2(4x + 3) < 50, where x is a positive, prime number.
b 2(3x – 1) ≤ 60, where x is a positive, square number.
c 4(5x – 3) ≤ 100, where x is positive but not a prime number.
3 Solve the following inequalities and illustrate their solutions on number lines.
a 5x – 4 < 11 b 3(2x + 5) ≤ 9
x > –1
x > –4
1 a On the same pair of axes, draw the graphs of the equations y = 2x + 1 and y = 2x + 3.
b Explain why there is no solution to this pair of simultaneous equations.
2 a Does every pair of linear simultaneous equations have a solution?
b Explain your answer to part a.
3 a Does every pair of simultaneous equations which do have a solution, have a unique solution?
b Explain your answer to part a.
4 Sketch a pair of graphs, one quadratic and one linear, which represent a pair of simultaneous
equations that will have only one solution.
CHAPTER 4 – GEOMETRY
1 Calculate the length of the hypotenuse in each of the following right-angled triangles.
Give your answers to one decimal place.
2 Calculate the length of the unknown side in each of the following right-angled triangles. Give your
answers to one decimal place
3 a Calculate x in the right-angled triangle shown on the right.
b Calculate the area of the triangle.
1 A plane flies due east for 120 km from airport A to airport B. It then flies due north for 280 km to
airport C. Finally, it flies directly back to airport A. Calculate the direct distance from airport C to airport
A. Give your answer to the nearest kilometer.
2 The length of a football pitch is 100 m and the width of the pitch is 80 m. Calculate the length of a
diagonal of the pitch. Give your answer to the nearest metre.
3 The regulations for the safe use of ladders states: For a 6 m ladder, the foot of the ladder must be
placed between 1.5 m and 2.2 m from the building.
a What is the minimum height the ladder can safely reach up the side of a building?
b What is the maximum height the ladder can safely reach up the side of a building?
4 Calculate the area of an equilateral triangle whose side length is 10 cm. Give your answer to one
decimal place.
1 Using a ruler and compasses, construct the locus which is equidistant from the points A and B.
2 Using a ruler and compasses, construct the locus which is equidistant from the perpendicular lines AB and BC.
3 Draw a diagram to show the locus of a set of points which are 4 cm or less from a fixed point X.
CHAPTER 5 – STATISTICS
1 The test results of ten pupils are recorded for four different subjects. Here are the results.
Pupil
French
Spanish
English
Music
A
45
52
63
35
B
64
60
56
45
C
22
30
46
58
D
75
80
70
30
E
47
60
55
42
F
15
24
40
50
G
80
74
68
42
H
55
65
53
48
I
85
77
75
41
J
33
47
51
50
a Plot the data for French and Spanish on a scatter graph. b Describe the relationship between French and
Spanish. c Plot the data for English and Music on a scatter graph. d Describe the relationship between
English and Music. e Plot the data for Spanish and English on a scatter graph. f Describe the relationship
between Spanish and English. g Use your answers to parts d and f to state the correlation between Music and
Spanish.
1 The table shows the scores of some pupils in a music exam and in a maths exam.
Pupil
A
B
C
D
E
F
G
H
I
J
Music
35
48
72
23
76
51
45
60
88
17
Maths
42
57
80
32
65
69
50
71
94
25
a Plot the data on a scatter graph. Use the x-axis for the music exam scores, from 0 to 100, and the y-axis for the
maths exam scores, from 0 to 100.
b Draw a line of best fit.
c One person did not do quite as well as expected on the maths test. Who do you think it was?
Give a reason.
2 A survey is carried out to compare the ages of people with the reaction time in a test.
Age (years) 45
83
24
76
63
44
42
37
50
62
Reaction
0.15 0.31 0.58 0.20 0.62 0.43 0.21 0.25 0.18 0.49
time
(seconds)
a Plot the data on a scatter graph. Use the x-axis for the range of ages, from 0 to 90 years, and the y-axis for
reaction times, from 0 to 1 seconds.
b Draw a line of best fit.
c Use your line of best fit to estimate the reaction time of a 30-year-old.
d Explain why it would not be sensible to use the line of best fit to predict the reaction time of someone aged
100.
1 Two fair spinners are spun and the scores are
a Compete the table of total scores.
added together to get a total score. This is
.
b List all the total scores which are prime numbers
recorded in the two-way table, shown below.
c State the most likely total scores
d Write down the probability of getting a total score of 7.
Give your answer as a fraction in its simplest form.
e Write down the probability of getting a total
score of 5. Give your answer as a fraction in
its simplest form
.
2 A year group recorded the days of the week on which they were born. Here are the results
a Write a comment on the births of boys and girls.
b Write a comment about the number of births on different days of the week
For each table of data:
a Copy and complete the cumulative frequency table.
b Draw the cumulative frequency graph.
c Use your graph to estimate the median and the interquartile range.
1 The height of 100 plants.
2
Height, h (cm)
0 < h ≤ 10
10 < h ≤ 20
20 < h ≤ 30
30 < h ≤ 40
40 < h ≤ 50
Number of plants
6
24
27
30
13
Height, h (cm) Cumulative frequency
h ≤ 10
h ≤ 20
h ≤ 30
h ≤ 40
h ≤ 50
The time that the school bus is late on 40 days.
Time, t (min)
Number of days
0<t≤5
5 < t ≤ 10
10 < t ≤ 15
15 < t ≤ 20
12
15
6
7
Time, t (min)
t≤5
t ≤ 10
t ≤ 15
t ≤ 20
Cumulative frequency
Copy and complete each table of values given below.
a Complete each table including the totals.
b Calculate an estimate of each mean.
Age, A (years)
11–12
13–14
15–16
17–18
Time, t, (hours)
0<t≤2
2<t≤4
4<t≤6
6<t≤8
Frequency, f
5
8
12
5
Total =
Mid-value, x, of age (years)
Frequency, f
2
7
10
5
Total =
12
f × x (years)
60
Mid-value, x, of time (hours)
1
f × x (hours
2
Total =
CHAPTER 6 – GEOMETRY
1 State whether each of the pairs of triangles below are similar.
2 a Explain why triangle ABC is similar to triangle PQR.
b Find the length of the side QR.
3 In the triangle below DE is parallel to BC. Find the length of BC
.
1
2
3
4
5
Express each of the following in mm2. a 3 cm2 b 8 cm2
c 4.5 cm2
d 0.8 cm2
2
2
2
Express each of the following in m . a 40 000 cm
b 70 000 cm c 32 000 cm2 d 5000 cm2
3
3
Express each of the following in cm . a 2 m
b 9 m3 c 3.7 m3 d 0.3 m3
Express each of the following in litres. a 8000 cm3 b 12 000 cm3 c 23 500 cm3 d 250 cm3
A rectangular park is 620 m long and 340 m wide. Find the area of the
park in hectares.
6 Calculate the volume of the box on the right. Give your answer in litres.
In this exercise take  = 3.142 or use the pi key on your
calculator.
1 Calculate: i the length of the arc and ii the area of the sector for each of the following circles.
Give your answers correct to three significant figures.
2
Calculate the area of the sectors below. Give your answer correct to three significant figures
In this exercise take π = 3.142 or use the π pi
key on your calculator.
1 Calculate the volume of each of the following cylinders. Give your answers correct to three
significant figures.
2 The diagram below shows a metal pipe of length 1 m. It has an internal diameter of 2.8 cm, and an
external diameter of 3.2 cm. Calculate the volume of metal in the pipe. Give your answer correct to
the nearest cubic centimetre.
3 A cylindrical can holds 2 litres of oil. If the height of the can is 25 cm, calculate the radius of the
base of the can. Give your answer correct to one decimal place.
1
2
3
4
5
Find the distance travelled by a hiker who walks for 3 hours at an average speed of 2.5 mph.
Find the time taken to drive a car 125 km at an average speed of 75 km/h.
A runner runs a 1000 m race in 3 minutes 20 seconds. Find his average speed in m/s.
Find the density of a gold ingot that has a mass of 4825 g and a volume of 250 cm3.
The density of sea water is 1.05 g/cm3. If a bucket with a capacity of 5 litres is filled with seawater,
find the mass of the water in the bucket. Give your answer in kilograms.
6 The density of cork is 0.25 g/cm3. Find the volume of a block of cork that has a mass of 120 g.
CHAPTER 7 - NUMBER
1 Write each of the following numbers in standard form.
a 63 000 000
b 0.000 74
c 322 000 d 83 300
e 0.000 000 71 f 92321
g 0.009 35 h 0.000 0005
2 Write each of the following standard form numbers as an ordinary number.
a 4.9 × 104
b 4.36 × 10-3
c 8.4 × 103 d 5.68 × 10-2
e 8 × 109
f 4.82 × 10-4
g 9.2 × 106 h 6.03 × 10-1
3 Write each of the following numbers in standard form.
a 68 × 103
b 37.8 × 10-5
c 0.87 × 10-3
d 58 × 10-4
1 Do not use a calculator for this question. Work out each of the following and give your answer in
standard form.
a (4 × 102) × (2 × 106)
b (5 × 103) × (4 × 102)
c (6 × 10–3) × (2 × 10–4)
d (9 × 10–2) × (3 × 108)
e (5 × 10–5) × (8 × 10–3) f (7 × 103) × (7 × 103)
2 You may use a calculator for this question. Work out each of the following and give your answer in
standard form. Do not round off your answers.
a 2.1 × 105) × (3.4 × 103)
b (3.2 × 103) × (1.5 × 104)
d (1.5 × 10–2) × (2.5 × 10–4)
e (3.8 × 10–4) × (2.8 × 104)
c (3.6 × 103) × (2.8 × 10–8)
f (8.6 × 104) × (1.5 × 10–7)
1 Do not use a calculator for this question. Work out each of the following and give your answer in standard
form.
a (8 × 105) ÷ (2 × 103)
b (4 × 105) ÷ (5 × 107)
c (6 × 103) ÷ (2 × 10-4)
d (1.2 × 10-3) ÷ (3 × 10-2)
e (6 × 106) ÷ (8 × 10-1)
f (5 × 102) ÷ (8 × 10-3)
2 You may use a calculator for this question. Work out each of the following and give your answer in
standard form. Do not round off your answers.
a (6.15 × 105) ÷ (1.5 × 102)
b (3.15 × 106) ÷ (1.4 × 10-1)
c (3.19 × 103) ÷ (1.45 × 10-2)
d (2.32 × 10-3) ÷ (2.9 × 10-5)
e (5.85 × 10-3) ÷ (6.5 × 103)
f (1.495 × 106) ÷ (4.6 × 10-2)
Do not use a calculator for Questions 1 and 2.
1 Find the upper and lower bounds between which the following quantities lie.
a In a hive there are 2000 bees to the nearest 100.
b The amount of honey in a jar is 200 ml to the nearest 10 ml.
c The width of a field is 70 m to the nearest metre.
d The mass of a loaf is 0.6 kg to the nearest 100 grams.
2 A poster is 2.5 metres by 1.5 metres, each measurement accurate to the nearest 10 cm.
a What are the upper and lower bounds for the length of the poster?
b What are the upper and lower bounds for the width of the poster?
c What are the upper and lower bounds for the perimeter of the poster?
3 A bottle of water holds 1 litre to the nearest centilitre.
a What is the smallest possible amount in the bottle?
b What is the greatest possible amount that 10 bottles could hold?
1 a = 10, b = 20 and c = 30. All values to the nearest whole number.
a Write down the upper and lower bounds of a, b and c.
b Work out the upper and lower bounds of each of the following.
i a×b
iii (a × b) + c iv c2
ii c ÷ a
2 A rectangle has an area of 120 cm2, measured to the nearest 10 cm2. The length is 15 cm,
measured
to the nearest cm.
a What is the greatest possible width of the rectangle?
b What is the least possible width of the rectangle?
1 Write each of the following fractions as a recurring decimal.
4
85
17
a 7
b 101
c 103
2 Write each of the following recurring decimals as a fraction in its simplest form.
a 0.54 b 0.246 c 0.2 d 0.12 e 0.37
8
iv c2
2 Use the power key and/or the cube/cube-root key on your calculator to work out each of these.
a 27
b 3.23
c
d
CHAPTER 8 – ALGEBRA
1 Simplify each of these:
a 5m × 4m b 9a × 3a c 4x × 5x × 6x d (4y)2
e 9z × 3z2 f
i
(2w)3 g 8m ÷ 2m h 18a3 ÷ 3a
15x6 ÷ 3x4 j
12y7 ÷ 3y2 k 10z6y3 + 2zy2 l
5w3 × 4w5 ÷ 2w4
2 Simplify each of the following:
a 2x3 × 4x7 b
12t6 ÷ 3t c 20m5 ÷ 5m3 d 3y × 2y5 e x-2 ÷ x-3
3 Simplify each of the following, leaving your answer in fraction form:
a x3 ÷ x5 b 4m2 ÷ m5 c 8x-4 ÷ 2x d 2x5 ÷ 3x8 e Ax × Bx-5 f Ax ÷ Bx-5
4 Write the following as fractions:
a 5–1 b 6–2 c 2–3 d x–2 e 2y–4 f
4z–5
1 Use a calculator to find the result to one decimal place.
a
b
46
c
31
d
74
e
129
215
2 Without a calculator, state the cube roots of each of the following numbers.
a 64
b 343
c 216
d 729
e 512
3 Find the cube root of each of the following. Give your answers to one decimal
place.
a 96
b 110
c 55
d 297
e 3000
4 State which, in each pair of numbers, is the larger. You may use a calculator.
a
20, 3 55
28, 3 149
b
c
18, 3 79
5 Write down the value of each of the following without using an index.
a 49
1
2
b 512
1
3
1
1
c 16 4
d 1024 5
1
e
(343) 3
1 A sledge sliding down a slope has travelled a distance, d metres, in time, t seconds, where
d = 5t + t2.
a Draw a graph to show the distances covered up to 6 seconds.
b Find the distance travelled after 3.8 seconds.
c Find the time taken to travel 50 metres.
2 The cost, C pence, for plating knives of length L cm, is given by the formula C = 50L + 7L2.
a Draw a graph to show the cost of plating knives up to 10 cm long.
b What would be the cost of plating a knife 8.7 cm long?
c What would be the length of a knife costing £4 to plate?
1 By drawing suitable graphs, solve this pair of simultaneous equations:
2x + y = 5
y = x3 - 1
There is only one solution.
2 The distance, d metres, a rocket is above the ground is given by
d = 2t + t3
where t is the time in seconds.
Draw the distance-time graph for the first 3 seconds.
CHAPTER 9 – STATISTICS
1
A builder is working on a patio. The probability that the weather is fine is 0.6, and the
probability that he has all the materials is 0.9. To complete the job in a day, he needs the
weather to be fine and to have all the materials.
a Draw a tree diagram to show all the possibilities. b Calculate the probability that he
completes the job in a day. c Calculate the probability that it is not fine and he does not
have all the materials.
2 A game is played three times. The probability of winning each time is 21 .
a Show that the probability of winning all three games is 81 . b What is the probability of
winning exactly one game?
1 Ten pictures are shown, which are all face down. A picture is picked at random.
a What is the probability of choosing a picture of a guitar?
b What is the probability of choosing a picture of a guitar or a boat?
c What is the probability of choosing a picture of a horse or a doll?
d What is the probability of choosing a picture which is not of a boat?
2 A bag contains a large number of discs, each labelled either A, B, C or D. The probabilities that
a disc picked at random will have a given letter are shown below.
P(A) = 0.2
P(B) = 0.4
P(C) = 0.15
P(D) = ?
a What is the probability of choosing a disc with a letter D on it?
b What is the probability of choosing a disc with a letter A or B on it?
c What is the probability of choosing a disc which does not have the letter C on it?
A spinner has different coloured sections. It is spun 100 times and the number of times it lands
on blue is recorded at regular intervals. The results are shown in the table.
a Copy and complete the table.
Number of spins
Number of times lands on blue
Relative frequency
20
40 60
80
100
6
10 15
22
26
0.3
b What is the best estimate of the probability of landing on blue?
c How many times would you expect the spinner to land on blue in 2000 spins?
d If there are two sectors of the spinner coloured blue, how many sectors do you think there
are altogether? Explain your answer.