Question

Each packet of washing powder carries a token and four tokens can be exchanged for a free packet. How many free packets will I receive if I buy 64 packets?

Answer 1

Hint: We can calculate the number of free packets by dividing the number of packets by the number of tokens providing free packets. Each free packet also comes with a token, collection of which ensures more free packets. The total number of free packets is given by calculating the sum of all.

Complete step-by-step answer:
Given information:
1 packet of washing powder = 1 token
1 free packet of washing powder = 4 tokens
Number of packets bought = 64
Now,
The number of free packets $\left( {{n_{{f_1}}}} \right)$ when 64 packet are bought is given by its division with number of tokens providing free packets which is given as:
\[
  {n_{{f_1}}} = \dfrac{{64}}{4} \\
  {n_{{f_1}}} = 16 \\
 \]
🡪 The number of free packets with 64 packets is 16.
With these 16 packets came 16 tokens, the number of free packets obtained by this:
\[
  {n_{{f_2}}} = \dfrac{{16}}{4} \\
  {n_{{f_2}}} = 4 \\
 \]
🡪 The number of free packets with 16 packets is 4.
And with these 4 packets come 4 tokens more, the number of free packets obtained by this:
\[
  {n_{{f_3}}} = \dfrac{4}{4} \\
  {n_{{f_3}}} = 1 \\
 \]
🡪 The number of free packets with 4 packets is 1
Only one token is obtained, so no more free packets.

The total number of free packets obtained are:
${n_f} = {n_{{f_1}}} + {n_{{f_2}}} + {n_{{f_3}}}$
Substituting the calculated values, we get:
$
  {n_f} = 16 + 4 + 1 \\
  {n_f} = 21 \\
 $
Therefore, I will receive 21 free packets if I buy 64 packets of washing powder.

Note: Taking tokens from free packets for more packets in consideration is an important step because the free packets are not only given by the total packets mentioned but also with the free packets already received on that.
If there is confusion between what quantity should be divided by whom, remember that in case of finding the numbers larger quantities will always be the numerator because the smaller in numerator might provide the number in decimal which is not countable for physical quantities like boxes, packets etc.