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Sometimes, rounding to one significant figure is not enough as this estimate may often be very far from the actual value.

In this question, you will explore some real life scenarios where estimating a value can be helpful. In these kinds of situations, it is often better to decide whether we are going to always round up or always round down, to ensure we don't go below/above the true value. Decide whether it is better to round up or round down and estimate the calculation:

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Round up.

", "

Round down.

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Yes, we do have enough cash to pay for this.

", "

No, we may not have enough cash to pay for this.

Imagine you are shopping at the supermarket. You only have £{cash} in cash. There are two items in your basket so far, costing £{p[0]} and £{p[1]}.

Just before checkout, you notice a tasty {flavour} ice cream on the shelf. It costs £{ice_cream}. Can you put this in your basket without going over your limit?

i) If we don't want to underestimate the total price of these 10 items, do we round the individual prices up, or down?

[[0]]

ii) Estimate the total price if we buy the ice cream, rounding the price of each item to 1 decimal place.

£ [[1]]

iii) Can we be sure that we have enough cash to pay for this?

[[2]]

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Overestimate and therefore we round up.

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Underestimate and therefore we round down.

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Imagine you just bought a new house. Your new bedroom's wall and ceiling are currently painted white, but you would like to paint these {colour}.

The dimensions of the floor are $\\var{length}\\,\\mathrm{m} \\times \\var{width}\\,\\mathrm{m}$ and the room is $\\var{height}\\,\\mathrm{m}$ high.

You need to find out how much paint to buy to paint all 4 walls and the ceiling.

i) Is it better to overestimate or underestimate in this situation?

[[0]]

ii) Rounding each measurement to the nearest metre, estimate the whole area to be painted {colour}.

[[1]] m²

iii) One bucket of {colour} paint is enough to paint an area of 15m². How many buckets should you buy to ensure you have enough paint?

[[2]]

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Overestimate and therefore we round up.

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Underestimate and therefore we round down.

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Imagine you are in the local gym, running on a treadmill. Your speed is $\\var{speed}$ m/s.

You don't have a calculator right now, but you want to estimate the distance you're going to run in five minutes.

i) Is it better to overestimate or underestimate now?

[[0]]

ii) What is the approximate distance you will run, rounding the speed to a whole number?

[[1]] m

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Distance $=$ Speed $\\times$ Time

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Students are asked to estimate values in possible real-life scenarios.

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a)

We want to ensure we won't go over the limit, so it is better to overestimate. If we underestimated, we could potentially think we have enough money while we don't.

To overestimate our total, we round up.

ii)

We round up all our values to 1 decimal place:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Original prices	$£\\var{p[0]}$	$£\\var{p[1]}$	$£\\var{ice_cream}$
Rounded up	$£\\var{precround(p[0]+0.05,1)}$	$£\\var{precround(p[1]+0.05,1)}$	$£\\var{precround(ice_cream,1)}$

Now we calculate the total of these rounded numbers:

\\[ £\\var{precround(p[0]+0.05,1)} + £\\var{precround(p[1]+0.05,1)} + £\\var{precround(ice_cream,1)} = £\\var{total_rounded_up} \\]

iii)

As the estimated total $£\\var{total_rounded_up}$ is higher than $£\\var{cash}$, we may not have enough money to purchase the {flavour} ice cream.

b)

It is much better to have residual paint than to not have enough of it. That is why it is better to overestimate the area.

Therefore, we round up.

ii)

We round up all our values to the nearest integer:

Length: $\\var{length}\\,\\mathrm{m} = \\var{l}\\,\\mathrm{m}$.

Width: $\\var{width}\\,\\mathrm{m} = \\var{w}\\,\\mathrm{m}$.

Height: $\\var{height}\\,\\mathrm{m} = \\var{h}\\,\\mathrm{m}$.

The total area consists of 3 areas:

Two walls of $\\var{l}\\,\\mathrm{m} \\times \\var{h}\\,\\mathrm{m}$ (length by height), two walls of $\\var{w}\\,\\mathrm{m} \\times \\var{h}\\,\\mathrm{m}$ (width by height) and a ceiling of $\\var{l}\\,\\mathrm{m} \\times \\var{w}\\,\\mathrm{m}$ (length by width).

\\[ \\begin{align}
\\var{l}\\,\\mathrm{m} \\times \\var{h}\\,\\mathrm{m} &= \\var{l*h}\\,\\mathrm{m}^2
\\\\ \\var{w}\\,\\mathrm{m} \\times \\var{h}\\,\\mathrm{m} &= \\var{w*h}\\,\\mathrm{m}^2
\\\\ \\var{l}\\,\\mathrm{m} \\times \\var{w}\\,\\mathrm{m} &= \\var{l*w}\\,\\mathrm{m}^2
\\end{align}\\]

Therefore, the total area we need to paint is

\\[ \\begin{align}
2 \\times \\var{l}\\,\\mathrm{m} \\times \\var{h}\\,\\mathrm{m} + 2 \\times \\var{w}\\,\\mathrm{m} \\times \\var{h}\\,\\mathrm{m} + \\var{l}\\,\\mathrm{m} \\times \\var{w}\\,\\mathrm{m} &= 2 \\times \\var{l*h}\\,\\mathrm{m}^2 + 2 \\times \\var{w*h}\\,\\mathrm{m}^2 + \\var{l*w}\\,\\mathrm{m}^2
\\\\ &= \\var{2*l*h}\\,\\mathrm{m}^2 + \\var{2*w*h}\\,\\mathrm{m}^2 + \\var{l*w}\\,\\mathrm{m}^2
\\\\ &= \\var{rall}\\,\\mathrm{m}^2
\\text{.} \\end{align}\\]

iii)

\\[\\var{rall}\\,\\mathrm{m}^2 \\div 15\\,\\mathrm{m}^2 = \\var{rall/15}\\]

We can only buy a whole number of buckets, so we need to decide between {buckets-1} and {buckets} paint buckets. As it is better to buy more paint than not buy enough, we should buy {buckets} buckets of {colour} paint.

c)

In this case, it is better to underestimate. It is less of an error to assume we ran less than the actual value. If we overestimated, we would assume we ran more than we actually did and would do less exercise than we initially wanted.

Therefore, we round down.

ii)

To calculate the distance we use the formula

\\[\\mathrm{Distance} = \\mathrm{Speed} \\times \\mathrm{Time}\\]

We round down the speed:

\\[\\var{speed} \\, \\mathrm{m/s} = \\var{rspeed} \\, \\mathrm{m/s} \\text{.}\\]

We convert time into seconds:

\\[\\begin{align} 5 \\, \\mathrm{min} &= 5 \\times 60 \\\\&= 300 \\, \\mathrm{s} \\text{.} \\end{align}\\]

Estimate distance:

\\[\\begin{align} \\mathrm{Distance} &= \\var{rspeed} \\, \\mathrm{m/s} \\times 300 \\, \\mathrm{s} \\\\&= \\var{dist} \\, \\mathrm{m} \\text{.} \\end{align}\\]

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