Power Play (Exponents) | FIO

Question 12

A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits 0–9, how many digits should the code consist of?

Solution

We need to find the smallest number of digits required to create a unique code for each milk packet.

Step 1 — Total number of packets

First, let us write the total number of milk packets in standard numerical form. One billion means $1,000,000,000$ . So, 8.5 billion is 8.5 multiplied by one billion.

$8.5 \text{ billion packets} = 8.5 \times 1,000,000,000 \text{ packets}$

$= 8,500,000,000 \text{ packets}$

We can also write this number using powers of 10. The number $1,000,000,000$ is equal to $10^9$ .

$8.5 \times 10^9 \text{ packets}$

$\boxed{8.5 \times 10^9 \text{ packets}}$

Step 2 — Codes possible with 'n' digits

We are using digits from 0 to 9. This means we have 10 different choices for each digit position in the code. If we have a code with one digit, we can make 10 unique codes (0, 1, 2, ..., 9). This is $10^1$ . If we have a code with two digits, we can make $10 \times 10 = 100$ unique codes (00, 01, ..., 99). This is $10^2$ . So, if we have a code with 'n' digits, we can make $10^n$ unique codes. Let 'n' be the number of digits in the code. The total number of unique codes we can make is $10^n$ . We need enough unique codes for all $8.5 \times 10^9$ packets. So, the number of possible codes must be greater than or equal to the number of packets.

$10^n \ge 8.5 \times 10^9$

Step 3 — Finding the minimum number of digits

We need to find the smallest whole number 'n' that satisfies the inequality $10^n \ge 8.5 \times 10^9$ . Let us try different whole number values for 'n'. If $n = 9$ , the number of unique codes is $10^9$ .

$10^9 = 1,000,000,000$

This is 1 billion. We need $8.5$ billion codes. Since $1,000,000,000$ is less than $8,500,000,000$ , 9 digits are not enough.

$10^9 < 8.5 \times 10^9$

Let us try the next whole number for 'n', which is 10. If $n = 10$ , the number of unique codes is $10^{10}$ .

$10^{10} = 10 \times 10^9$

$= 10 \times 1,000,000,000$

$= 10,000,000,000$

This is 10 billion. Since $10,000,000,000$ is greater than $8,500,000,000$ , 10 digits are enough.

$10^{10} \ge 8.5 \times 10^9$

So, the smallest number of digits needed is 10.

$\boxed{\text{10 digits}}$

Answer

The code must consist of 10 digits.

More questions in FIO

Express the following in exponential form:

(i) $6 \times 6 \times 6 \times 6$

(ii) $y \times y$

(iii) $b \times b \times b \times b$

(iv) $5 \times 5 \times 7 \times 7 \times 7$

(v) $2 \times 2 \times a \times a$

(vi) $a \times a \times a \times c \times c \times c \times c \times d$

Express each of the following as a product of powers of their prime factors in exponential form:

(i) 648

(ii) 405

(iii) 540

(iv) 3600

Write the numerical value of each of the following:

(i) $2 \times 10^3$

(ii) $7^2 \times 2^3$

(iii) $3 \times 4^4$

(iv) $(-3)^2 \times (-5)^2$

(v) $3^2 \times 10^4$

(vi) $(-2)^5 \times (-10)^6$

Find out the units digit in the value of $2^{224} \div 4^{32}$ ? [Hint: $4 = 2^2$ ]

There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would there be after 40 days?

Write the given number as the product of two or more powers in three different ways. The powers can be any integers.

(i) $64^3$ (ii) $192^8$ (iii) $32^{-5}$

Examine each statement below and find out if it is 'Always True', 'Only Sometimes True', or 'Never True'. Explain your reasoning.

(i) Cube numbers are also square numbers.

(ii) Fourth powers are also square numbers.

(iii) The fifth power of a number is divisible by the cube of that number.

(iv) The product of two cube numbers is a cube number.

(v) $q^{46}$ is both a 4th power and a 6th power ( $q$ is a prime number).

Simplify and write these in the exponential form.

(i) $10^{-2} \times 10^{-5}$

(ii) $5^7 \div 5^4$

(iii) $9^{-7} \div 9^4$

(iv) $(13^{-2})^{-3}$

(v) $m^5n^{12}(mn)^9$

If $12^2 = 144$ what is

(i) $(1.2)^2$

(ii) $(0.12)^2$

(iii) $(0.012)^2$

(iv) $120^2$

Q10

Circle the numbers that are the same—

$2^4 \times 3^6$ $6^4 \times 3^2$ $6^{10}$ $18^2 \times 6^2$ $6^{24}$

Q11

Identify the greater number in each of the following—

(i) $4^3$ or $3^4$

(ii) $2^8$ or $8^2$

(iii) $100^2$ or $2^{100}$

Q12

Q13

64 is a square number ( $8^2$ ) and a cube number ( $4^3$ ). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Q14

A digital locker has an alphanumeric (it can have both digits and letters) passcode of length 5. Some example codes are G89P0, 38098, BRJKW, and 003AZ. How many such codes are possible?

Q15

The worldwide population of sheep (2024) is about $10^9$ , and that of goats is also about the same. What is the total population of sheep and goats?

(ii) $20^9$

(ii) $10^{11}$

(iii) $10^{10}$

(iv) $10^{18}$

(v) $2 \times 10^9$

(vi) $10^9 + 10^9$

Q16

Calculate and write the answer in scientific notation:

(i) If each person in the world had 30 pieces of clothing, find the total number of pieces of clothing.

(ii) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees.

(iii) The human body has about 38 trillion bacterial cells. Find the bacterial population residing in all humans in the world.

(iv) Total time spent eating in a lifetime in seconds.

Q17

What was the date 1 arab/1 billion seconds ago?