Rounding, Estimation & Bounds (Edexcel IGCSE Maths A: Higher): Exam Questions

The value of $p$ is 4.3
The value of $q$ is 0.4

Both $p$ and $q$ are given correct to the nearest $0.1$

Write down the lower bound for $p$.

      $r=p+\frac{1}{q}$

Work out the upper bound for $r$.
You must show all your working.

$I=5(v—u)$

$$\begin{gathered} v=14\mathrm{correct}\mathrm{to}2\mathrm{significant}\mathrm{figures} \\ u=8.7\mathrm{correct}\mathrm{to}2\mathrm{significant}\mathrm{figures} \end{gathered}$$

Work out the upper bound for the value of $I$.
You must show your working.

$a=\frac{v-u}{t}$

$\begin{array}{rcl}v& =& 37.6\mathrm{correct}\mathrm{to}3\mathrm{significant}\mathrm{figures}.\\ u& =& 11.3\mathrm{correct}\mathrm{to}3\mathrm{significant}\mathrm{figures}.\\ t& =& 8.4\mathrm{correct}\mathrm{to}2\mathrm{significant}\mathrm{figures}.\end{array}$

Work out the upper bound for the value of $a$.
Show your working clearly.

$D=\frac{x}{y}$

$$\begin{gathered} x=99.7\mathrm{correct}\mathrm{to}1\mathrm{decimal}\mathrm{place}. \\ y=67\mathrm{correct}\mathrm{to}2\mathrm{significant}\mathrm{figures}. \end{gathered}$$

Work out an upper bound for $D$.

There are $892$litres of oil in Mr Aston's oil tank.
He uses $18.7$litres of oil each day.

Estimate the number of days it will take him to use all the oil in the tank.

Work out an estimate for $\frac{31\times 9.87}{0.509}$

Work out an estimate for the value of  $\sqrt{63.5\times 101.7}$.

${(2.3)}^6=148$correct to $3$ significant figures.

Find the value of ${(0.23)}^6$ correct to $3$ significant figures.

Show that $5^{-2}=\frac{1}{25}$

$k=\frac{t}{a-h}$

$t$= 14 correct to 2 significant figures
$a$ = 7.8 correct to 2 significant figures
$h$ = 3.4 correct to 2 significant figures

Work out the lower bound for the value of $k$.
Show your working clearly.

Nav has worked out $\frac{68.3\times 42.8}{0.021}$ on his calculator.

His answer is $139201.9048$

Without using a calculator and using suitable approximations, check that his answer is sensible.
Show your working clearly.

$P=ef$

$\begin{array}{rcl}e& =& 4.8\mathrm{correct}\mathrm{to}2\mathrm{significant}\mathrm{figures}.\\ f& =& 0.26\mathrm{correct}\mathrm{to}2\mathrm{significant}\mathrm{figures}.\end{array}$

Work out the lower bound for the value of $P$.
Show your working clearly.
Give your answer correct to $3$ significant figures.

$Q=\frac{t}{w}$
$\begin{array}{rcl}t& =& 2.73\mathrm{correct}\mathrm{to}3\mathrm{significant}\mathrm{figures}.\\ w& =& 0.04\mathrm{correct}\mathrm{to}1\mathrm{significant}\mathrm{figure}.\end{array}$

Work out the upper bound for the value of $Q$.
Show your working clearly.
Give your answer correct to $2$ significant figures.

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To the nearest pound, Jon has £9

To the nearest 50p, Ellie has £6.50

Work out the maximum possible total amount of money.

£........................................ 

A solid metal cube has sides of length 216 mm, correct to 3 significant figures. The cube is melted down and the metal used to make solid spheres.

The volume of each sphere is to be 120 cm³, correct to the nearest 10 cm³

Work out the least number of spheres that can be made from the metal. Show your working clearly.

Dan does an experiment to find the value of $\pi .$
He measures the circumference and the diameter of a circle.

He measures the circumference, $C,$ as $170$ mm to the nearest millimetre.
He measures the diameter, $d,$ as $54$ mm to the nearest millimetre.

Dan uses $\pi =\frac{C}{d}$ to find the value of $\pi$.
Calculate the upper bound and the lower bound for Dan's value of $\pi .$

Steve travelled from Ashton to Barnfield.

He travelled $235$miles, correct to the nearest $5$ miles.    
The journey took him $200$ minutes, correct to the nearest $5$ minutes.

Calculate the lower bound for the average speed of the journey.
Give your answer in miles per hour, correct to $3$ significant figures.    
You must show all your working.

Jarek uses the formula

$\mathrm{Area}=\frac{1}{2}ab\sin C$

to work out the area of a triangle.

For this triangle,

    $$\begin{gathered} a=7.8\mathrm{cm}\mathrm{correct}\mathrm{to}\mathrm{the}\mathrm{nearest}\mathrm{mm}. \\ b=5.2\mathrm{cm}\mathrm{correct}\mathrm{to}\mathrm{the}\mathrm{nearest}\mathrm{mm}. \\ C={63}^{\circ }\mathrm{correct}\mathrm{to}\mathrm{the}\mathrm{nearest}\mathrm{degree}. \end{gathered}$$

Calculate the lower bound for the area of the triangle.

$a$ is 8.3 cm correct to the nearest mm
$b$ is 6.1 cm correct to the nearest mm

Calculate the upper bound for $c$.
You must show your working.

A train travelled along a track in 110 minutes, correct to the nearest 5 minutes.

Jake finds out that the track is 270 km long.
He assumes that the track has been measured correct to the nearest 10 km.

Could the average speed of the train have been greater than 160 km/h?
You must show how you get your answer.

Jake's assumption was wrong.
The track was measured correct to the nearest 5 km.

Explain how this could affect your decision in part (a).

The petrol consumption of a car, in litres per 100 kilometres, is given by the formula

         $\mathrm{Petrol}\mathrm{consumption}=\frac{100\times \mathrm{Number}\mathrm{of}\mathrm{litres}\mathrm{of}\mathrm{petrol}\mathrm{used}}{\mathrm{Number}\mathrm{of}\mathrm{kilometres}\mathrm{travelled}}$

Nathan's car travelled 148 kilometres, correct to 3 significant figures. The car used 11.8 litres of petrol, correct to 3 significant figures.

Nathan says,

   "My car used less than 8 litres of petrol per 100 kilometres."

Could Nathan be wrong?
You must show how you get your answer.

A cone has a volume of 98 cm³.
The radius of the cone is 5.13 cm.

Work out an estimate for the height of the cone.

John uses a calculator to work out the height of the cone to 2 decimal places.

Will your estimate be more than John's answer or less than John's answer?
Give reasons for your answer.

Work out an estimate for $\sqrt{4.98+2.16\times 7.35}$

Margaret has some goats.
The goats produce an average total of 21.7 litres of milk per day for 280 days.
Margaret sells the milk in $\frac{1}{2}$ litre bottles.

Work out an estimate for the total number of bottles that Margaret will be able to fill with the milk.
You must show clearly how you got your estimate.

In a competition, a prize is won every 2014 seconds.

Work out an estimate for the number of prizes won in 24 hours. You must show your working.

The mass of Jupiter is 1.899 x 10²⁷ kg.
The mass of Saturn is 0.3 times the mass of Jupiter.

Work out an estimate for the mass of Saturn.
Give your answer in standard form.

Give evidence to show whether your answer to (a) is an underestimate or an overestimate.

$D=\frac{u^2}{2a}$

$\begin{array}{rcl}u& =& 26.2\mathrm{correct}\mathrm{to}3\mathrm{significant}\mathrm{figures}\\ a& =& 4.3\mathrm{correct}\mathrm{to}2\mathrm{significant}\mathrm{figures}\end{array}$

Calculate the upper bound for the value of $D$.
Give your answer correct to $6$ significant figures.
You must show all your working.

The lower bound for the value of $D$ is $78.6003$ correct to $6$ significant figures.

By considering bounds, write down the value of $D$ to a suitable degree of accuracy.
You must give a reason for your answer.

Edith’s van can safely carry a maximum load of 920 kilograms.

She wants to use her van to carry

30 sacks of potatoes, each of mass 25 kilograms to the nearest kilogram

and

20 sacks of carrots, each of mass 7.5 kilograms to 1 decimal place.

Can she definitely use her van safely in one journey?

You must show your working.

The length of a roll of ribbon is 30 metres, correct to the nearest half-metre.

A piece of length 5.8 metres, correct to the nearest 10 centimetres, is cut from the roll.

Work out the maximum possible length of ribbon left on the roll.

...................metres

Claudine cycled a distance of 53 km in 2.7 hours.
The distance is measured correct to the nearest km.
The time is given correct to 1 decimal place.

Calculate the lower and upper bounds of her average speed.

Give your answers correct to 2 decimal places.

Lower bound = ................................ km/h     
Upper bound = ..................................... km/h

Sunil makes 7.5 litres of soup, correct to the nearest 0.5 litre.
He serves the soup in 300 ml portions, correct to the nearest 10 ml. 24 people order this soup.

Does Sunil definitely have enough soup to serve the 24 people? Show how you decide.

A solid metal cube has sides of length 125 mm, correct to 3 significant figures.

The cube is melted down and the metal used to make solid spheres.
The volume of each sphere is to be 140 cm³, correct to the nearest 10 cm³

Work out the greatest number of spheres that could be made from the metal.
Show your working clearly

A solid sphere has a radius of 2.8 centimetres, correct to 1 decimal place.
The sphere has a mass of $M\pi$ grams, where $M=260$ correct to 2 significant figures.

Work out the upper bound for the density of the sphere.
Give your answer in g / cm³ correct to 2 decimal places. Show your working clearly.

$P=a\left(c+y\right)$

$a=$8.3 correct to 2 significant figures

$c=$2 correct to 1 significant figure

$y=$15 correct to the nearest 5

Work out the upper bound for the value of $P$

Show your working clearly

$m=\frac{\sqrt{s}}{t}$

$$\begin{gathered} s=3.47\mathrm{correct}\mathrm{to}2\mathrm{decimal}\mathrm{places} \\ t=8.132\mathrm{correct}\mathrm{to}3\mathrm{decimal}\mathrm{places} \end{gathered}$$

By considering bounds, work out the value of $m$ to a suitable degree of accuracy.
You must show all your working and give a reason for your final answer.

A road is 4530 m long, correct to the nearest 10 metres.   
Kirsty drove along the road in 205 seconds, correct to the nearest 5 seconds.

The average speed limit for the road is 80 km/h.

Could Kirsty's average speed have been greater than 80 km/h?    You must show your working.

Jackson is trying to find the density, in g/cm³, of a block of wood. The block of wood is in the shape of a cuboid.

He measures    
the length as 13.2 cm, correct to the nearest mm    
the width as 16.0 cm, correct to the nearest mm    
the height as 21.7 cm, correct to the nearest mm

He measures the mass as 1970 g, correct to the nearest 5 g.

By considering bounds, work out the density of the wood.
Give your answer to a suitable degree of accuracy.

You must show all your working and give a reason for your final answer.

Sanders has a water tank for storing rainwater. 

The tank is in the shape of a cylinder.
The radius of the cylinder is 31 cm.
The height of the cylinder is 97.5 cm.

The tank is full of water.

Work out an estimate for the volume of water in the tank.
Give your answer in litres.
You must show your working.

Use 1000 cm³ = 1 litre.

A cycle race across America is 3069.25 miles in length.

Juan knows his average speed for his previous races is 15.12 miles per hour.
For the next race across America he will cycle for 8 hours per day.

Estimate how many days Juan will take to complete the race.

Juan trains for the race.
The average speed he can cycle at increases.
It is now 16.27 miles per hour.

How does this affect your answer to part (a)?

One uranium atom has a mass of $3.95\times {10}^{-22}$ grams.

Work out an estimate for the number of uranium atoms in 1 kg of uranium.

Is your answer to (a) an underestimate or an overestimate?

Give a reason for your answer.

The dimensions of a rectangular floor are to the nearest 0.1 metres.

A force of 345 Newtons is applied to the floor.
The force is to the nearest 5 Newtons.

Work out the upper bound of the pressure.
Give your answer to 4 significant figures.
You must show your working.

........................N/m² [5]

A £1 coin weighs 8.75 g, correct to the nearest 0.01 g. Mitul weighs the contents of a large bag of £1 coins. The coins weigh 2.63 kg, correct to the nearest 10 g.

Mitul says

I am sure that the bag contains exactly £300 because, using bounds,
   2625 ÷ 8.755 = 299.8 to 1 decimal place.

Show that Mitul may not be correct.

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