Sunday Times Teaser 2499 – No Title

by Victor Bryant

Published August 14 2010 (link)

Joe was talking to the supermarket manager about shelf stocking, and she mentioned a recent exercise with milk. Each morning, they added sufficient new stock to ensure there were 1,200 litres on the shelves. They found that 56% of the new stock was sold on the first day, half that amount the next day and half that new amount the day after. Any remaining stock was regarded as out of date and removed next morning before restocking. Hence, they calculated the number of litres to be added each morning.

What is that number?

One Comment Leave one →

Brian Gladman permalink

Assume \(M\) litres of milk is added each day

Sold amounts:
fresh (\(56 M / 100\))
1 day old (\(28 M / 100\))
2 days old (\(14 M / 100\))

Left over amounts:
fresh \((100 – 56) M / 100\) = \(44 M / 100\)
1 day old \((100 – 56 – 28) M / 100\) = \(16 M / 100\)
2 days old \((100 – 56 – 28 – 14) M / 100\) = \(2 M / 100\)

So to make up 1200 litres (2 day old milk is discarded):

\(M + 44 M / 100 + 16 M / 100 = 1200\)

Hence \(M\) is 750 litres.

This can be confirmed with a simulation:


# the amount of two day old milk
a = 0
# the amount of one day old milk
b = 0
# the amount of fresh milk
c = 0
# the amount of milk added each day
x = 1000
for i in range(30):
  # update at the start of each day
  a, b, c = b, c, x
  print(f'{a:.2f}, {b:.2f}, {c:.2f}')
  # 56% of the milk added per day is sold today
  c -= 0.56 * x
  # 28% is sold yesterday
  b -= 0.28 * x
  # 14% is sold two days ago
  a -= 0.14 * x  
  # two day old milk (a) is now discarded so the
  # amount needed to make up 1200 litres is:
  x = 1200 - b - c
print(f'The amount added each day is {x:.0f} litres.')

# the amount of two day old milk

a = 0

# the amount of one day old milk

b = 0

# the amount of fresh milk

c = 0

# the amount of milk added each day

x = 1000

for i in range(30):

# update at the start of each day

a, b, c = b, c, x

print(f'{a:.2f}, {b:.2f}, {c:.2f}')

# 56% of the milk added per day is sold today

c -= 0.56 * x

# 28% is sold yesterday

b -= 0.28 * x

# 14% is sold two days ago

a -= 0.14 * x

# two day old milk (a) is now discarded so the

# amount needed to make up 1200 litres is:

x = 1200 - b - c

print(f'The amount added each day is {x:.0f} litres.')

Sunday Times Teaser 2499 – No Title

by Victor Bryant

Published August 14 2010 (link)

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