Multiplication and division
Problems involving multiplication and division
When solving problems involving upper and lower bounds we have to:
- identify what the question is asking us to calculate
- find either the upper or lower bounds of the measurements
- work out your problem using the correct bounds
- ask yourself if your answer makes sense?
Multiplication
Example
A square tile is 18.5 cm long measured to the nearest mm.
- Write down the smallest possible length and the largest possible length of the tile.
- Tegan places three of these tiles across paths which are 55 cm wide measured to the nearest cm. Will the tiles always cover the path?
Solution
- Smallest possible length of the tile = 18.45. Largest possible length of the tile = 18.55.
- For the tiles to always cover the path, three of the smallest tiles must cover the biggest width of the path.
Smallest possible length of three tiles = 3 × 18.45 = 55.35 cm.
Largest possible width of the path = 55.5 cm.
No, the tiles will not always cover the width of the path.
Question
A bucket has a volume of 200 cm3, measured to the nearest 10 cm3.
- Write down the smallest and largest possible values of the volume of the bucket.
- Water is poured from the bucket into a paddling pool of volume 10.5 litres measured to the nearest 0.1 litre. Is it always possible to pour water from 50 full buckets into the tank without it overflowing?
- Smallest possible volume of the bucket = 195 cm3. Largest possible volume of the bucket = 205 cm3.
- To ensure the water never overfills the paddling pool, you need to see if the maximum amount of water fits into the smallest paddling pool.
50 of largest buckets = 50 × 205 = 10,250 cm3.
Smallest capacity of paddling pool = 10.45 litres.
There are 1,000 cm3 in a litre.
1 l = 1,000 cm3.
10.45 l = 10.45 × 1,000 = 10,450 cm3.
The capacity of the pool is larger than the amount of water from the buckets. This means the paddling pool will not overflow.
Division
Example
Usain Bolt set a new world record for sprinting the 200 m at the 2009 World Championships. The official time was 19.19 s but we’ll round this up to 19.2 s (one decimal place).
The distance is measured to the nearest m and the time is measured to the nearest tenth of a second. Find the fastest and slowest speed of Usain Bolt in metres per second.
Solution
\(Speed = \frac {Distance} {Time}\)
Maximum Time = 19.25 s
Minimum Time = 19.15 s
Maximum Distance = 200.5 m
Minimum Distance = 199.5 m
\(Speed = \frac {Distance} {Time}\)
For the greatest value when you divide, you need the largest distance divided by the shortest time.
Fastest speed = \(\frac {200.5} {19.15}\) = 10.46997389 m/s (or ms-1)
For the smallest value when you divide, you need the smallest value divided by the largest.
Slowest speed = \(\frac {199.5} {19.25}\) = 10.36363636 m/s (or ms-1)
Question
The mass of a piece of metal is measured as 5 kg to the nearest kg. The volume of the metal is 130 cm3 correct to the nearest 10 cm3. Calculate the maximum possible density of the metal.
The formula for \({Density}~=~\frac {Mass} {Volume}\)
For the maximum density, divide the largest mass by the smallest volume.
Maximum density = \(\frac {5.5} {125}\) = 0.044 kg/cm3
