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7.2D Bounds: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculator allowed · about 65 minutes

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Bounds: Fluency and Exam Drill

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This worksheet builds fluency in finding upper and lower bounds from values that have been rounded or truncated, before applying bounds to combined calculations such as density, average speed and formulae with more than one rounded quantity. The final two questions stretch towards grade 9, asking you to reason backwards from a calculated bound and to explain the asymmetry in error interval notation.

The length of a straight running track is marked as 80 m, correct to the nearest 10 m.
Write down the lower bound and the upper bound of the exact length.

(Total for Question 1 is 2 marks)

The temperature inside a greenhouse is recorded as 8 degC, correct to the nearest whole number.
Write down the error interval for the exact temperature, t.

(Total for Question 2 is 2 marks)

The distance of a charity fun run is given as 12.4 km, correct to 1 decimal place.
Find the upper and lower bounds of the exact distance.

(Total for Question 3 is 2 marks)

The attendance at an outdoor concert in Cardiff is reported as 2400, correct to the nearest 100.
Find the upper and lower bounds for the exact attendance.

(Total for Question 4 is 2 marks)

A bag of plain flour weighs 4.7 kg, correct to 1 decimal place.
Write down the error interval for the exact weight, w, of the flour.

(Total for Question 5 is 2 marks)

A gold ring has a mass of 5.28 g, correct to 2 decimal places.
Find the upper and lower bounds of its exact mass.

(Total for Question 6 is 2 marks)

The wingspan of a model glider is given as 70 cm, correct to 1 significant figure.
Find the upper and lower bounds of the exact wingspan.

(Total for Question 7 is 2 marks)

The concentration of a saline solution is recorded as 0.036 mol/dm³, correct to 2 significant figures.
Find the upper and lower bounds of the exact concentration.

(Total for Question 8 is 2 marks)

The length of a garden cane, p, is measured as 3.9 m after truncating (rounding down) to 1 decimal place.
Write down the error interval for the exact length, p.

(Total for Question 9 is 2 marks)

Two offcuts of skirting board have lengths a = 14 cm and b = 9 cm, each measured correct to the nearest cm.
Find the upper bound of a + b.

(Total for Question 10 is 1 mark)

Using the same two offcuts (a = 14 cm, b = 9 cm, each correct to the nearest cm), find the lower bound of a - b.

(Total for Question 11 is 2 marks)

p = 6.2 and q = 3.4, each correct to 1 decimal place.
Find the upper bound of p * q.

(Total for Question 12 is 2 marks)

r = 50, correct to the nearest 10, and s = 8, correct to the nearest whole number.
Find the lower bound of r / s.

(Total for Question 13 is 2 marks)

w = 4.5, correct to 1 decimal place.
Find the lower bound of w^2.

(Total for Question 14 is 2 marks)

A bottle of cooking oil has a mass of 460 g, correct to the nearest 10 g, and a volume of 500 cm³, correct to the nearest 10 cm^3.
Density = mass / volume.

(a)Find the upper bound of the density of the oil. Give your answer correct to 3 significant figures.(3)

(b)Find the lower bound of the density of the oil. Give your answer correct to 3 significant figures.(2)

(Total for Question 15 is 5 marks)

A runner completes a half marathon distance of 21 km, correct to the nearest km, in a time of 1.8 hours, correct to the nearest 0.1 hour.

(a)Calculate the upper bound for the average speed of the runner. Give your answer correct to 3 significant figures.(3)

(b)A coach claims the runner's average speed 'must have been at least 12 km/h'. By finding the lower bound of the average speed, determine whether this claim can be guaranteed.(2)

(Total for Question 16 is 5 marks)

A rectangular vegetable bed has length 6.3 m and width 2.8 m, each measured correct to 1 decimal place.

(a)Find the upper bound of the perimeter of the bed.(2)

(b)Find the lower bound of the area of the bed.(2)

(Total for Question 17 is 4 marks)

A cylindrical water butt has a base radius of 25 cm, correct to the nearest cm, and a height of 90 cm, correct to the nearest 5 cm.
Volume of a cylinder = π * r² * h.
Calculate the upper bound for the volume of the water butt. Give your answer correct to 3 significant figures.

(Total for Question 18 is 3 marks)

T = (a - b) / c, where a = 15.6, b = 8.2 and c = 3.5, each correct to 1 decimal place.
Find the lower bound of T. Give your answer correct to 3 significant figures.

(Total for Question 19 is 3 marks)

Four identical lengths of curtain track are joined end to end to make a track spanning a bay window. Each length of track measures 1.8 m, correct to 1 decimal place.
A fitter claims the total span could be as much as 7.4 m.
Comment on whether the fitter's claim is accurate, using bounds to justify your answer.

(Total for Question 20 is 3 marks)

The area of a rectangular allotment plot is calculated, using bounds, to satisfy 47.25 ≤ A < 47.95 (A in m²). One side of the plot, x, is known exactly to be 7 m.

(a)Find the lower bound and the upper bound of the length of the other side, y, of the plot.(2)

(b)State the degree of accuracy to which y must originally have been measured (rounded) to produce these bounds, giving a reason.(2)

(Total for Question 21 is 4 marks)

A length L is recorded as 6.0 cm, correct to 1 decimal place, so its error interval is written as 5.95 ≤ L < 6.05.
Ben says: 'The interval should be 5.95 ≤ L ≤ 6.05, because 6.05 rounds to 6.0 in the same way that 5.95 does.'
Explain why Ben is wrong, and confirm the correct error interval for L.

(Total for Question 22 is 3 marks)

Mark scheme · 7.2D Bounds: Fluency and Exam Drill

Question 1

B1 75 (m) cao
B1 85 (m) cao
Answer: Lower bound = 75 m, upper bound = 85 m

Question 2

B1 7.5 ≤ t oe
B1 t < 8.5 oe
Answer: 7.5 ≤ t < 8.5

Question 3

B1 12.45 (km) cao
B1 12.35 (km) cao
Answer: Upper bound = 12.45 km, lower bound = 12.35 km

Question 4

B1 2450 cao
B1 2350 cao
Answer: Upper bound = 2450, lower bound = 2350

Question 5

B1 4.65 ≤ w oe
B1 w < 4.75 oe
Answer: 4.65 ≤ w < 4.75

Question 6

B1 5.285 (g) cao
B1 5.275 (g) cao
Answer: Upper bound = 5.285 g, lower bound = 5.275 g

Question 7

B1 75 (cm) cao
B1 65 (cm) cao
Answer: Upper bound = 75 cm, lower bound = 65 cm

Question 8

B1 0.0365 (mol/dm³) cao
B1 0.0355 (mol/dm³) cao
Answer: Upper bound = 0.0365 mol/dm³, lower bound = 0.0355 mol/dm³

Question 9

B1 3.9 ≤ p oe
B1 p < 4.0 oe
Answer: 3.9 ≤ p < 4.0

Question 10

B1 24 (cm) cao
Answer: 24 cm

Question 11

M1 lower bound of a = 13.5 and upper bound of b = 9.5 oe
A1 4 (cm) cao
Answer: 4 cm

Question 12

M1 upper bound of p = 6.25 and upper bound of q = 3.45 oe
A1 21.5625 oe, awrt 21.6
Answer: 21.5625 (awrt 21.6)

Question 13

M1 lower bound of r = 45 and upper bound of s = 8.5 oe
A1 5.29 awrt (3 sf)
Answer: 5.29 (awrt 3 sf)

Question 14

M1 lower bound of w = 4.45 oe
A1 19.8025 oe, awrt 19.8
Answer: 19.8025 (awrt 19.8)

Question 15

(a) M1 upper bound of mass = 465 and lower bound of volume = 495 oe
(a) M1 dep 465 / 495 oe
(a) A1 0.939 (g/cm³) awrt
(a) Answer: 0.939 g/cm³ (awrt)
(b) M1 lower bound of mass = 455 and upper bound of volume = 505, used as 455 / 505 oe
(b) A1 0.901 (g/cm³) awrt
(b) Answer: 0.901 g/cm³ (awrt)

Question 16

(a) M1 upper bound of distance = 21.5 oe
(a) M1 dep lower bound of time = 1.75, used as 21.5 / 1.75 oe
(a) A1 12.3 (km/h) awrt
(a) Answer: 12.3 km/h (awrt)
(b) M1 finds lower bound of speed = 20.5 / 1.85 (= 11.081...) oe
(b) A1 correct conclusion with reason, e.g. No - the lower bound (awrt 11.1 km/h) is less than 12 km/h, so the speed could have been below 12 km/h, meaning the claim cannot be guaranteed
(b) Answer: No - the average speed could have been as low as 11.1 km/h (lower bound), so the claim that it must have been at least 12 km/h cannot be guaranteed.

Question 17

(a) M1 upper bound of length = 6.35 and upper bound of width = 2.85, used in 2(length + width) oe
(a) A1 18.4 (m) cao
(a) Answer: 18.4 m
(b) M1 lower bound of length = 6.25 and lower bound of width = 2.75, used in length x width oe
(b) A1 17.1875 (m²) oe, awrt 17.2
(b) Answer: 17.1875 m² (awrt 17.2)

Question 18

M1 upper bound of radius = 25.5 and upper bound of height = 92.5 oe
M1 dep π x 25.5² x 92.5 oe
A1 189000 (cm³) awrt 3 sf
Answer: 189000 cm³ (awrt 3 sf)

Question 19

M1 lower bound of a = 15.55 and upper bound of b = 8.25 oe
M1 dep upper bound of c = 3.55, used as (15.55 - 8.25) / 3.55 oe
A1 2.06 awrt (3 sf)
Answer: 2.06 (awrt 3 sf)

Question 20

M1 upper bound of one length = 1.85 oe
M1 dep 4 x 1.85 (= 7.4) oe
A1 correct conclusion with supporting reasoning, e.g. the upper bound of the total span is exactly 7.4 m, but since this bound is not itself included in the interval, the true span must be strictly less than 7.4 m, so it could get arbitrarily close to, but never actually reach, 7.4 m
Answer: The maximum possible span (upper bound) is 7.4 m, but this value is not included in the error interval, so the true span must be less than 7.4 m - it can get close to 7.4 m but never equal or exceed it.

Question 21

(a) M1 divides both ends of the area interval by 7, e.g. 47.25 / 7 and 47.95 / 7 oe
(a) A1 6.75 ≤ y < 6.85 oe, both bounds correct
(a) Answer: 6.75 ≤ y < 6.85
(b) B1 recognises the interval 6.75 to 6.85 has a half-width of 0.05, matching rounding to 1 decimal place oe
(b) B1 states y was measured correct to 1 decimal place (nearest 0.1 m), consistent with a recorded value of 6.8 m
(b) Answer: y was measured correct to 1 decimal place (to the nearest 0.1 m); the recorded value would have been 6.8 m.

Question 22

B1 correctly shows or states that 6.05 rounds (to 1 dp) up to 6.1, not down to 6.0, since the digit after the first decimal place is 5 oe
B1 correctly explains that 5.95 does round to 6.0 (to 1 dp), which is why the lower bound is included with ≤ while the upper bound, which would round to 6.1, is excluded with < oe
B1 confirms the correct error interval 5.95 ≤ L < 6.05 oe
Answer: Ben is wrong: 6.05 rounds to 6.1 (not 6.0), so it cannot be included in the interval for a value that rounds to 6.0, while 5.95 does round to 6.0, so it is included. The correct error interval is 5.95 ≤ L < 6.05.

Bounds: Fluency and Exam Drill - Worksheets, Questions and Revision