DAY 90

Comment

Another way of putting this is to ask whether the gap between 1/6 and 1/7 is greater or smaller than the gap between 1/7 and 1/8.

 

Answer

1/7 is nearer 1/8 than 1/6.

 

Readers' methods

Stella D, Skipton BD23
(also, Lillian G, Perth):
Halfway between 1/6 and 1/8 is (1/6 + 1/8)/2 = 7/48.
1/7 = 7/49, which is smaller than 7/48, so 1/7 is nearer to 1/8.

Michael S, Sunderland SR4
(similar methods from M Coveney, London N15; David M, Hereford HR1; Stephen Ginns; Sue M, Hemel Hempstead; Terry R, Bracknell RG12; David Y, Cottingham HU16; P Gould, Halifax HX2, P Wrtentmore, Swansea SA2; Martin Flavell):
As fractions become smaller so the difference between them approaches zero.
In going from the largest to the smallest, 1/7 is nearer 1/8 than 1/6.

S Hope, Northampton NN7
(similar methods from John B, Homerton; P Wrentmore, Swansea SA2):
The problem is equivalent to
Is 1/2 nearer 1 than 1/3 ?
Or, Is 3/6 nearer 6/6 or 2/6 ?

Mark B, Helston TR12:
Consider some similar but more extreme questions.
Which is closer out of these: 1/2 and 1/3 or 1/99 and 1/100 ?
Obviously 1/99 and 1/100 are much closer than 1/2 and 1/3.
So the differences between fractions decrease as the denominator increases.
Therefore 1/7 is closer to 1/8 than to 1/6.

C Harris, Bristol BS5
(similar methods from Michael S, Sunderland SR4; JulieW, Sandbach; Paul A, Newcastle NE6; Alma V, Seehaven; Michael B, Canterbury CT1; Isabel H, Co Down BT33; P Wrentmore, Swansea SA2; Michael J, Croydon CR0; Michelle R, London SE23):
Find a common denominator.
Multiply 7 by 6 by 8 = 336.
1/7 = 48/336, 1/6 = 56/336, 1/8 = 42/336.
So the difference between 1/7 and 1/6 is 8/336,
and the difference between 1/7 and 1/8 is 6/336.
So 1/7 is nearer 1/8.

Andrew B, Norfolk NR17
(similar methods from Martin Flavell; Chris F, London NW10; John C, Camborne TR14):
The common denominator is 6x7x8. There is no need to work this out.
 

 1/6

  

 1/7

  

 1/8

 numerators

 7x8

  

 6x8

  

 6x7

 

 56

  

 48

  

 42

 differences  

 ­8

  

 ­6

  

Now it is clear that 1/7 is closer to 1/8 than 1/6.

Debbie C, Penzance TR20
(similar methods from Alan L, Eastbourne BN23; Roy G, Norwich NR2; Jonathan J, S Croydon CR2; Charles H, West Horsley, KT24; Stephen Ginn; Marta de V H, Granada, Spain; Jeanette S, Orkney KW15; A Galsworthy, Dudley DY3; Peter R, Sawbridgeworth CM21; J McCrellis, Riseley MK44):
Break 6, 7, 8 into factors:
7 = 7, 6 = 2x3, 8 = 2x2x2.
Lowest common multiple = 7x2x3x2x2 = 168.
1/7 = (2x2x3x2)/168 = 24/168
1/8 = (7x3)/168 = 21/168
1/6 = (7x2x2)/168 = 28/168.
Therefore 1/7 is nearer 1/8.

Roy G, Norwich NR2
(similar methods from John B, Homerton; Stephen Fahey; David M, Hereford HR1; Mark B, Helston TR12; J McGrellis, Riseley MK44; Susan T, Wisbech):
Look at the 'gaps' between the numbers.
1/7 ­ 1/8 = (8­7)/56 = 1/56
1/6 ­ 1/7 = (7­6)/42 = 1/42.
As 1/56 is less than 1/42, there is a smaller gap between 1/8 and 1/7 than between 1/6 and 1/7.

Niall O'Connor, Co Cork, Ireland:
The difference between two successive fractions like 1/2 and 1/3 is 1/(2x3), ie 1/6.
From inspection it is clear that 1/(7x8) is smaller than 1/(6x7), and hence 1/7 is closer to 1/8 than 1/6.

John B, Homerton
(similar methods from Roy G, Norwich NR2; Jeanette S, Orkney KW15):
1/(n+1) ­ 1/(n+2) = 1/(n+1)(n+2)
1/n ­ 1/(n+1) = 1/n(n+1).
(n+1)(n+2) > n(n+1), so 1/(n+1)(n+2) < 1/n(n+1).

Michael S, Sunderland SR4
(similar methods from Andrew S, Powys LD1; Yerni A, London SW15; Chrissie B, Bristol BS6; K Hewitt, Flintshire CH7; Roy G, Norwich NR2; Andrew B, Norfolk NR17; M Coveney, London N15; Stephen Ginns; Mark B, Helston TR12; C Polley, Swindon SN4; Marta de V H, Granada, Spain; Jeanette S, Orkney KW15; Michelle R, London SE23; J McGrellis, Riseley MK44):
The decimal method (to 4 places) shows
1/6 = 0.1667, 1/7 = 0.1429, 1/8 = 0.1250.
Thus 1/6 ­ 1/7 is 0.0238 and 1/7 ­ 1/8 is 0.0179,
therefore 1/7 is closer to 1/8 by 0.0059.

Jane H, Leicester LE9
I would use a number line and display it visually.
The number line could be 336 cm long (a common multiple of 6, 7, 8).
(Similar methods from Mark B, Helston TR12 (168 mm line); Jan C, Bristol BS14 (168 unit squares); Margaret W, Stone ST15 (10 cm line); C Sharp, Manchester M41 (336 unit squares); John R, Dringhouses YO 24 (6 cm x7 cm x 8 cm cuboid))

L E G Whitehead, Twickenham TW1
(similar method from Sue M, Hemel Hempstead):
Common multiple = 6x7x8 = 336.
Find 1/6 of 336 = 56.
Find 1/7 of 336 = 48.
Find 1/8 of 336 = 42.
Therefore 1/7 is nearer 1/8 than 1/6.
(Similar method using 168 from E Gould, Halifax HX2; Sophia P, London N8)

S M Coates, St Gregory's School:
Divide 100 by 6 = 16.666666
Divide 100 by 7 = 14.285714
Divide 100 by 8 = 12.5.
The difference between 1/6 and 1/7 is 2.380952
The difference between 1/7 and 1/8 is 1.785714.
Therefore the seventh is closer tothe eighth.

Jill T, Bournemouth BH8:
1/8 of 8" = 1"
1/7 of 8" = 1.14"
1/6 of 8" = 1.33".
Therefore smaller difference between 1/7 and 1/8.

Jane H, Ashtead KT21
(similar methods from Vic B, Leicester LE3; Cathy R, Alvechurch B48; J McGrellis, Riseley MK44):
Draw three circles, one divided into six equal parts, one divided into 7 equal parts, and the third divided into 8 equal parts.
How many degrees for the 6 ? Answer 60 degrees.
How many degrees for the 7 ? Answer 51.4 degrees.
How many degrees for the 8 ? Answer 45 degrees.
51.4 is only 6.4 bigger than 45 while 60 degrees is 8.6 bigger than 51.4, so one seventh is nearer to one eighth.

Cathy R, Alvechurch B48:
1/7 of a week is 1 day.
1/6 of a week is 1/6 x 7 x 24 hours = 28 hours.
1/8 of a week is 1/8 x 7 x 24 hours = 21 hours.
So 1/7 is closer to 1/8.

Eric C, London SW17:
The problem is equivalent to
Is 7/7 nearer to 7/6 or 7/8 ?