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# Estimating

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## Estimating

You should always do a quick estimate in your head when doing arithmetic so you can see if your answer is reasonable.

**Sometimes an exam question will test your ability to do this!**

Generally, you should round each number involved to **one** significant figure (see **Rounding** Learn-it) and then it's easy to estimate by using the single digits and moving the point around.

**Let's have a look at one:**

**936 x 27** - this is difficult to do in your head but if we round both numbers to one significant figure it becomes **900 x 30.**

Now this is easy to do in your head by doing **9 x 3 = 27** then moving the point 3 times (putting three noughts on!) giving the answer **27 000** which is a good estimate of the real answer **25 272.**

**Here's some more!**

**45 x 72** becomes **50 x 70** which is **3500**

**317 x 23** becomes **300 x 20** which is **6000**

Check these are reasonable estimates of the real answers!

**Here's a more difficult one; click the boxes to reveal the estimates:**

**Tip!**

If you can't perform your estimate in your head then it's too complicated - **think again!**

It is important to remember that most measurement is **approximate.**

If you say your garden is 8 metres long you are rounding to the nearest metre and it could be anything from 7.5 to 8.5 metres long.

**Upper and Lower Bounds**

The real value can be as much as **half the rounded unit** above or below the value given.

So, if you are given 5.4cm the **upper bound** is 5.45cm and the **lower bound** is 5.35cm.

For 6.0kg you need to go 0.05kg either way so the **upper bound** is 6.05kg and the **lower bound** is 5.95kg.

**Maximum and Minimum Values**

For calculations you must use the upper or lower bounds of each measurement depending on what calculation you are doing.

**Addition** - For the maximum use the upper bound of each measurement, for the minimum use the lower bound of each measurement.

**For example:**

If a piece of wood measuring 15cm is joined to another piece measuring 12cm you can see the maximum and minimum values of the addition by clicking below.

**Subtraction** - For the maximum you need the biggest difference between the two measurements i.e. the upper bound of the first number and the lower bound of the second and for the minimum it's the other way round.

**For example:**

David and Steven were given seeds to plant in Biology and decided to see whose would grow the highest. After two weeks they measured them to the nearest centimetre and David's had grown to 11cm whereas Steven's had grown to 15cm. **What are the maximum and minimum values of Steven's victory?**

**Multiplication** - Same as for Addition

**Division** - Same as for Subtraction

**Tip!**

If it is a complicated calculation e.g. (32.3 x 42.6) - 12.7 then remember the rules for each separate operation. For a **maximum** this would be (32.35 x 42.65) - 12.65 (Notice the **lower bound** was used for 12.7 as it was a subtraction).

Sometimes we have to find the answer to something by simply guessing! We may then try another guess to see if it is better and so on until we are happy with our answer.

This is called **Trial and Improvement** and there are two main rules:

**1. ** Use tables to display your guesses and the answer they gave.

**2.** Be methodical. Don't guess randomly!

**For example:**

a rectangle has an area of 100cm^{2 }and its base is 1cm more than it's height. Find its height to 2 decimal places.

**Start with whole numbers.**

Height (cm) |
Base (cm) |
Area (cm^{2}) |

9 | 10 | 90 |

10 | 11 | 110 |

Now we know the height must be between 9 and 10 so we move on to the first decimal place. We can start with 9.5 if we want!

Height (cm) |
Base (cm) |
Area (cm^{2}) |

9.5 | 10.5 | 99.75 (too low!) |

9.6 | 10.6 | 101.76 (too high!) |

Now we know the height is between 9.5 and 9.6 so we can move to the second decimal place.

Height (cm) |
Base (cm) |
Area (cm^{2}) |

9.55 | 10.55 | 100.7525 (too high!) |

9.54 | 10.54 | 100.5516 (too high!) |

9.53 | 10.53 | 100.3509 (too high!) |

9.52 | 10.52 | 100.1504 (too high!) |

9.51 | 10.51 | 99.9501 (too low!) |

Now we know it's between 9.51 and 9.52. We only want two decimal places so the answer has to be one of these!

We try the middle i.e. 9.515. The answer is **too high** so we now know it is nearer to **9.51**

To two decimal places the height is **9.51cm.**

**Try to guess Tom's height:**

**Tom's is somwhere between 100cm and 200cm tall. Enter your guess in the box and click on the 'GO' button.**

**After each guess you can see if it is too low or too high, use this to narrow down your guesses.**

**See how many tries it takes you.**