Find the smudged value

The idea

Since mean = sum of values ÷ number of values, the total equals mean × number of values even when the data is not given; a missing value is this total minus the known values.

Coach Balwan's smudge

Coach Balwan wrote down the weights of 10 kushti wrestlers and their mean, 39.2 kg. Then water smudged one weight. Nine weights can still be read.

Can the missing weight be found?

• No. It is smudged, so it is lost.
• Yes. The mean still tells us something.
• Only by weighing the wrestler again.

Find w

The top bar shows the total if every wrestler weighed exactly the mean. The bottom bar puts the known weights end to end and ends with the smudged weight w. Slide w until the bottom bar is exactly as long as the top bar. Do both lists.

Total = mean × number of values

Since mean = sum of values ÷ number of values, the total equals mean × number of values even when the data is not given; a missing value is this total minus the known values.

Balwan: ten wrestlers, mean 39.2. The total must be 39.2 × 10 = 392 The nine known weights add up to 349, so w = 392 − 349 = 43 The missing weight is 43 kg.

The book writes it as an equation: (42 + 40 + … + 32 + w) ÷ 10 = 39.2, which simplifies to 349 + w = 392.

Venkayya's coconuts: the mean is 25.6 per tree and there are 15 trees. We do not have the 15 counts, but the total is 25.6 × 15 = 384 One tree was counted 3 too many, so the true total is 384 − 3 = 381 and the true mean is 381 ÷ 15 = 25.4

Notes

Since mean = sum of values ÷ number of values, the total equals mean × number of values even when the data is not given; a missing value is this total minus the known values.

Check yourself

The mean of 5 numbers is 12. Four of them are 8, 10, 14 and 11. What is the fifth number?

Answer: 17

The total of five numbers is 5 × 12 = 60. The four known numbers add up to 8 + 10 + 14 + 11 = 43. The fifth number is 60 − 43 = 17.

Venkayya's mean harvest is 25.6 coconuts per tree for 15 trees. One tree's count was written 3 more than the real number. What is the correct mean?

Answer: 25.4

The total was 25.6 × 15 = 384. The true total is 384 − 3 = 381, so the true mean is 381 ÷ 15 = 25.4.

The mean of 8 numbers is 15, but you cannot see the numbers. What do you know for sure?

The mean of 5 numbers is 18. Two of them are 12 and 21. The other three are all equal. What is each of the three?

Answer: 19

The total of five numbers is 5 × 18 = 90. Take away 12 + 21 = 33: 57 is left for three equal numbers, so each is 57 ÷ 3 = 19.

• Every number is 15.. No: 10 and 20 have mean 15 and neither of them is 15. Only the total is fixed.
• Their total is 8 + 15 = 23.. Adding the count and the mean means nothing. The mean is total ÷ count, so the total is mean × count.
• Their total is 8 × 15 = 120. — correct. Yes! The mean is the total ÷ 8, so the total is 15 × 8 = 120, even though we cannot see the numbers.
• Their total is 15 ÷ 8.. That divides the wrong way. The mean is the total ÷ the count, so the total is the mean × the count.