[discia] GCSE Maths › Number › Bounds

Bounds

discia.org 10 questions · 39 marks
Name Date
  1. Calculator 3 marks

    Ben writes down three measurements.

    The length of a table is 260 cm, correct to the nearest 10 cm.

    The mass of a book is 1.2 kg, correct to 1 decimal place.

    The capacity of a bottle is 850 ml, correct to the nearest 50 ml.

    (a)

    Write down the upper bound of the length of the table.

    (b)

    Write down the lower bound of the mass of the book.

    (c)

    Write down the upper bound of the capacity of the bottle.

    Show answer
    (a) \(265\ \text{cm}\)
    (b) \(1.15\ \text{kg}\)
    (c) \(875\ \text{ml}\)
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  2. Calculator 3 marks

    The mass of a chocolate bar is 65 g, correct to the nearest 5 g.

    (a)

    Could the true mass have been exactly 66.75 g?

    Explain your answer.

    (b)

    Could the true mass have been exactly 62.5 g?

    Explain your answer.

    Show answer
    (a) Yes — the true mass satisfies \(62.5 \leq x < 67.5\) g, and 66.75 sits inside that interval.
    (b) Yes — the true mass satisfies \(62.5 \leq x < 67.5\) g, and 62.5 sits exactly at the lower bound, which the interval does include.
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  3. Calculator Higher3 marks

    Lengths of skirting board are laid end to end.

    Each one has a length of 210 cm, correct to the nearest 10 cm.

    There are exactly 6 of them, and they are all identical.

    (a)

    Calculate the upper bound of the total length.

    (b)

    Calculate the lower bound of the total length.

    Show answer
    (a) \(1290\ \text{cm}\)
    (b) \(1230\ \text{cm}\)
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  4. Calculator 3 marks

    \(k\) is a whole number between 5 and 35.

    (a)

    The square root of \(k\) is 5, to the nearest whole number.

    Work out the smallest possible value of \(k\).

    (b)

    Grace says that the smallest possible value of \(k\) is \(16\).

    Give a reason why Grace is not correct.

    Show answer
    (a) \(21\)
    (b) \(16\) is less than \(20.25\), so its square root is less than \(4.5\) and rounds to \(4\), not \(5\).
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  5. Calculator Higher3 marks

    To be approved, a playground must have an area of at least 7689 m\(^2\).

    The playground is a rectangle measuring 120 m by 80 m, each correct to the nearest 20 m.

    Show that the playground can be approved.

    Show answer
    Lower bounds are \(110\) m and \(70\) m, so the smallest possible area is \(110 \times 70 = 7700\) m\(^2\). Since 7700 m\(^2\) is greater than 7689 m\(^2\), the playground has enough area whatever the true sizes are, so it can be approved.
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  6. Calculator Higher6 marks

    Amara weighs two parcels.

    Parcel A has a mass of 8.4 kg, correct to 1 decimal place.

    Parcel B has a mass of 3.1 kg, correct to 1 decimal place.

    (a)

    Find the upper bound of the total mass of the two parcels.

    (b)

    Find the upper bound of the difference between the two masses.

    (c)

    Find the lower bound of the difference between the two masses.

    Show answer
    (a) \(11.6\ \text{kg}\)
    (b) \(5.4\ \text{kg}\)
    (c) \(5.2\ \text{kg}\)
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  7. Calculator Higher6 marks

    A rectangular sheet of card has a length of 30 cm and a width of 25 cm.

    Both measurements are correct to the nearest centimetre.

    (a)

    Calculate the upper bound of the area of the sheet of card.

    (b)

    Calculate the lower bound of the area of the sheet of card.

    Show answer
    (a) \(777.75\ \text{cm}^2\)
    (b) \(722.75\ \text{cm}^2\)
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  8. Calculator Higher3 marks

    Tomas is loading a van with furniture.

    The van can safely carry 1386 kg.

    Tomas has already loaded 1250 kg of furniture into the van, correct to the nearest 50 kg.

    One more crate has a mass of 90 kg, correct to the nearest 10 kg.

    Can the crate definitely be added to the van without the total mass going over 1386 kg? You must show your working.

    Show answer
    Yes — the load already in could be up to \(1250 + 25 = 1275\) kg, and the crate could be up to \(90 + 5 = 95\) kg, so the total could be up to \(1275 + 95 = 1370\) kg, which is within the 1386 kg limit, so the crate can definitely be added safely.
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  9. Calculator Higher4 marks

    This sign shows when a lift is safe to use.

    0100200Maximum 143 kgTotal mass of people the lift can safely carry (kg)

    Anya and some other people are in the lift. Their total mass is 210 kg, correct to the nearest 10 kg.

    Anya gets out. Anya has a mass of 73 kg, correct to the nearest kg.

    Is the lift now safe to use? You must show your working.

    Show answer
    Yes. The limit is 143 kg. The remaining mass is at most \(215 - 72.5 = 142.5\) kg and at least \(205 - 73.5 = 131.5\) kg, so the greatest possible remaining mass is not over the limit.
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  10. Calculator Higher5 marks

    A remote-controlled car travels 460 m, correct to the nearest 10 m.

    It takes 76.8 seconds, correct to the nearest 0.1 seconds.

    \(\text{average speed} = \text{distance} \div \text{time}\)

    (a)

    Find the upper bound of the average speed.

    Give your answer in m/s correct to 3 significant figures.

    (b)

    Find the lower bound of the average speed.

    Give your answer in m/s correct to 3 significant figures.

    Show answer
    (a) \(6.06\ \text{m/s}\)
    (b) \(5.92\ \text{m/s}\)
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discia.org Bounds · sheet 1
Answers Bounds · sheet 1
  1. (a) \(265\ \text{cm}\)
    (b) \(1.15\ \text{kg}\)
    (c) \(875\ \text{ml}\)
  2. (a) Yes — the true mass satisfies \(62.5 \leq x < 67.5\) g, and 66.75 sits inside that interval.
    (b) Yes — the true mass satisfies \(62.5 \leq x < 67.5\) g, and 62.5 sits exactly at the lower bound, which the interval does include.
  3. (a) \(1290\ \text{cm}\)
    (b) \(1230\ \text{cm}\)
  4. (a) \(21\)
    (b) \(16\) is less than \(20.25\), so its square root is less than \(4.5\) and rounds to \(4\), not \(5\).
  5. Lower bounds are \(110\) m and \(70\) m, so the smallest possible area is \(110 \times 70 = 7700\) m\(^2\). Since 7700 m\(^2\) is greater than 7689 m\(^2\), the playground has enough area whatever the true sizes are, so it can be approved.
  6. (a) \(11.6\ \text{kg}\)
    (b) \(5.4\ \text{kg}\)
    (c) \(5.2\ \text{kg}\)
  7. (a) \(777.75\ \text{cm}^2\)
    (b) \(722.75\ \text{cm}^2\)
  8. Yes — the load already in could be up to \(1250 + 25 = 1275\) kg, and the crate could be up to \(90 + 5 = 95\) kg, so the total could be up to \(1275 + 95 = 1370\) kg, which is within the 1386 kg limit, so the crate can definitely be added safely.
  9. Yes. The limit is 143 kg. The remaining mass is at most \(215 - 72.5 = 142.5\) kg and at least \(205 - 73.5 = 131.5\) kg, so the greatest possible remaining mass is not over the limit.
  10. (a) \(6.06\ \text{m/s}\)
    (b) \(5.92\ \text{m/s}\)
discia.org Bounds · sheet 1