Real Numbers, Surds & Rounding (Grade 10) — Worksheet & Memo

∑ DANEMATHICS

FREE CAPS MATHS RESOURCES

Grade 10 · Algebraic Expressions

Real Numbers, Surds & Rounding (Grade 10)

MARKING GUIDELINE

Marks

Duration

45 minutes

Questions

Name:

Class:

Date:

Mark

/ 30

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
√112 lies between which TWO integers?
(1)
A)9 and 10
B)10 and 11✓
C)11 and 12
D)55 and 56
Answer: B — 10² = 100 and 11² = 121, and 112 sits between them.
- A — 9² = 81, which is already well below 112
- C — 11² = 121 is ABOVE 112, so 11 is the upper integer, not the lower
- D — halved 112 instead of square-rooting it
1.2
Which ONE of these numbers is IRRATIONAL?
(1)
A)16
B)20✓
C)0,25
D)227
Answer: B — √20 = 4,472… — it never terminates and never recurs.
- A — √16 = 4, a whole number
- C — a terminating decimal is rational
- D — a ratio of two integers is rational by definition
1.3
Rounded to THREE decimal places, 3,04528 is …
(1)
A)3,046
B)3,045✓
C)3,05
D)3,0453
Answer: B — The fourth decimal is 2, which is less than 5, so the third decimal stays a 5.
- A — rounded using the 8 further along instead of the very next digit
- C — that is TWO decimal places
- D — that is FOUR decimal places
1.4
Rounded to THREE significant figures, 0,004936 is …
(1)
A)0,004
B)0,00494✓
C)0,005
D)0,00493
Answer: B — Leading zeros are not significant, so the first three significant digits are 4, 9 and 3; the next digit is 6, so the 3 rounds up.
- A — counted the leading zeros as significant figures
- C — that is ONE significant figure
- D — did not round the third digit up
1.5
Which ONE of these is a NON-TERMINATING, RECURRING decimal?
(1)
A)18
B)13✓
C)2
D)0,75
Answer: B — 13 = 0,333… — the 3 repeats forever, so it is rational but never terminates.
- A — 1 ÷ 8 = 0,125, which TERMINATES
- C — √2 never recurs at all — it is irrational
- D — 0,75 terminates after two decimals
1.6
−3 belongs to which ONE of these sets?
(1)
A)the natural numbers
B)the integers✓
C)the whole numbers
D)the irrational numbers
Answer: B — The integers are the whole numbers together with their negatives.
- A — the natural numbers start at 1 and include no negatives
- C — the whole numbers start at 0 and include no negatives
- D — −3 is a ratio of integers, so it is RATIONAL
1.7
Which statement about √−9 is true?
(1)
A)it equals −3
B)it is non-real✓
C)it equals 3
D)it is irrational
Answer: B — No real number multiplied by itself gives a negative, so √−9 is non-real.
- A — (−3)² = 9, not −9
- C — 3² = 9, not −9
- D — irrational numbers are still REAL numbers
1.8
A plank 3,7 m long is cut into pieces 0,8 m long. How many WHOLE pieces are cut?
(1)
A)5
B)4✓
C)4,625
D)4,6
Answer: B — 3,7 ÷ 0,8 = 4,625, and only whole pieces can be cut, so 4.
- A — rounded 4,625 up, but the fifth piece cannot be cut from what is left
- C — that is the unrounded quotient, not a number of pieces
- D — rounded to one decimal place instead of to a whole piece
1.9
Without a calculator, the best estimate of √40 is …
(1)
A)20
B)6,3✓
C)4,5
D)8
Answer: B — 6² = 36 and 7² = 49, so √40 sits just above 6; indeed 6,3² = 39,69.
- A — halved 40 instead of square-rooting it
- C — 4,5² = 20,25, nowhere near 40
- D — 8² = 64, which overshoots 40
1.10
A circle of radius 5 cm has area 78,539816… cm². Correct to TWO decimal places this is …
(1)
A)78,53 cm²
B)78,54 cm²✓
C)78,5 cm²
D)79 cm²
Answer: B — The third decimal is 9, so the second decimal 3 rounds up to 4.
- A — truncated instead of rounding — the next digit is 9
- C — that is ONE decimal place
- D — that is rounded to the nearest whole number

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[9 MARKS]

Answer the following questions WITHOUT the use of a calculator.

2.1
Determine between which TWO consecutive integers 112 lies.
(3)
10² = 100 and 11² = 121 (2)
112 lies between them, so 10 < 112 < 11 (1)
2.2
Consider the list: −7 ; 16 ; 5 ; 227 ; π ; 0,272727…
Write down ALL the IRRATIONAL numbers from the list.
(3)
5 and π (2)
The rest are rational: 16 = 4 and 0,272727… = 311 (1)
2.3
Explain why 227 is RATIONAL even though it is used as an approximation for the irrational number π.
(3)
It is a ratio of two integers (1), so by definition it is rational (1)
Its decimal recurs, while π never repeats — the two numbers are not equal (1)

Question 3

[11 MARKS]

Answer the questions below.

3.1
Round 3,14159 off to THREE decimal places, and 0,04567 off to THREE decimal places.
(3)
3,14159 → 3,142 (1)
0,04567 → 0,046 (2)
3.2
Determine, without a calculator, between which two consecutive integers 54 lies, and state which of the two 54 is CLOSER to. Justify your answer.
(5)
7² = 49 and 8² = 64 (2)
7 < 54 < 8 (1)
54 is 5 above 49 but 10 below 64 (1)
So it is closer to 7 (1)
3.3
A learner rounds 12,996 to TWO decimal places and writes 12,99. Identify the error and give the correct answer.
(3)
The learner truncated instead of rounding (1)
The third decimal is 6, so the second must round UP (1)
13,00 (1)

TOTAL: 30 marks

This question paper consists of 3 questions.