Binary multiples and file size calculations

What you'll learn

The storage units Edexcel expects: bit, nibble, byte, KiB, MiB, GiB and TiB.
Why GCSE Computer Science uses binary multiples based on 1024, not decimal thousands.
How to convert between storage units accurately.
How to construct expressions for file size and total data capacity.

Starting point: data is stored as bits

Computers store data using binary, a number system that uses only two digits: 0 and 1. Each 0 or 1 is a bit.

Definition

Bit

A bit is a single binary digit: either 0 or 1. It is the smallest unit of data storage.

Bits are grouped into larger units so that file sizes are easier to describe. It would be awkward to say a small image is 2,457,600 bits every time, so we often convert into bytes, KiB or MiB.

The storage units you need

A nibble is 4 bits. A byte is 8 bits, or 2 nibbles.

After bytes, Edexcel uses binary multiples:

Unit	Symbol	Meaning
bit	b	one binary digit
nibble	—	4 bits
byte	B	8 bits
kibibyte	KiB	1024 bytes
mebibyte	MiB	1024 KiB
gibibyte	GiB	1024 MiB
tebibyte	TiB	1024 GiB

Definition

Binary multiple

A binary multiple is a storage unit based on powers of 2. In this topic, each larger unit from KiB upwards is 1024 times the previous unit because 1024 is 2 to the power of 10.

This ladder shows the direction of the conversions.

Storage unit conversion ladder from bits to tebibytes

Key Idea

The unit ladder

Move towards smaller units by multiplying. Move towards larger units by dividing. From byte upwards, the factor is 1024 each time.

Common Mistake

Using 1000 instead of 1024

For this Edexcel GCSE topic, use KiB, MiB, GiB and TiB, not decimal kB/MB/GB/TB. So 1 KiB is 1024 bytes, not 1000 bytes.

Bits and bytes: the capital letter matters

The symbol b usually means bit. The symbol B means byte.

That tiny difference changes the size by a factor of 8.

64 b means 64 bits.
64 B means 64 bytes, which is 512 bits.

Tip

Sanity check

If a file size is in bits, divide by 8 to get bytes. If it is in bytes, multiply by 8 to get bits.

Example

Converting bits to bytes

A small data item is 2048 bits. Convert this to bytes.

Use the relationship 1 byte = 8 bits, so the number of bytes is found by dividing the number of bits by 8.
Substitute the value:
$bytes=2048÷8\text{bytes} = 2048 \div 8$
Calculate the result:
$bytes=256\text{bytes} = 256$
So 2048 bits is 256 B.

Converting between KiB, MiB, GiB and TiB

From bytes upwards, each step uses 1024.

So:

bytes to KiB: divide by 1024
KiB to MiB: divide by 1024
MiB to GiB: divide by 1024
GiB to TiB: divide by 1024

Going the other way, multiply by 1024 each step.

Example

Converting MiB to bytes and bits

Convert 3 MiB into bytes and then into bits.

Convert MiB to KiB by multiplying by 1024:
$\times 1024 = 3072 \text{ KiB}$
Convert KiB to bytes by multiplying by 1024 again:
$3072 \times 1024 = 3,145,728 \text{ B}$
Convert bytes to bits by multiplying by 8:
$\times 8 = 25,165,824 \text{ bits}$
So 3 MiB is 3,145,728 B, or 25,165,824 bits.

File size: build the expression first

A file size is the amount of storage needed to store a file.

The main GCSE skill is not just “doing sums”; it is choosing the correct expression from the information given.

Key Idea

General file size pattern

A file size calculation usually follows this pattern: number of items multiplied by bits needed for each item. Then convert the answer into the unit requested.

For example:

For text, the items are characters. A character is one symbol such as A, 7, ? or a space.
For a bitmap image, the items are pixels. A pixel is one tiny coloured square in the image. The colour depth is the number of bits used for each pixel.
For sound, the items are samples. A sample is one measurement of the sound wave. The sample rate tells you how many samples are recorded per second, and the bit depth tells you how many bits are used per sample.

You may not need all of these formulae in one question, but the same structure appears again and again.

Common expressions

For text:

character\text{file size in bits} = \text{number of characters} \times \text{bits per character}

For a bitmap image:

depth\text{file size in bits} = \text{width} \times \text{height} \times \text{colour depth}

For sound:

depth\text{file size in bits} = \text{sample rate} \times \text{duration} \times \text{bit depth}

If the question gives multiple channels, such as stereo sound, include that too:

depth×channels\text{file size in bits} = \text{sample rate} \times \text{duration} \times \text{bit depth} \times \text{channels}

Common Mistake

Only include what the question gives you

GCSE file size questions usually ignore extra storage such as file headers or metadata unless the question explicitly tells you to include it.

Example

Calculating a bitmap image file size

An uncompressed bitmap image is 640 pixels wide and 480 pixels high. Each pixel uses 8 bits. Calculate the file size in KiB.

Construct the expression in bits using width, height and colour depth:
$size=640×480×8\text{file size} = 640 \times 480 \times 8$
Calculate the number of bits:
$640 \times 480 \times 8 = 2,457,600 \text{ bits}$
Convert bits to bytes by dividing by 8:
$\div 8 = 307,200 \text{ B}$
Convert bytes to KiB by dividing by 1024:
$\div 1024 = 300 \text{ KiB}$
The file size is 300 KiB.

Data capacity requirements

A data capacity requirement is the total amount of storage needed for a system, device or collection of files.

This often means calculating the size of one file, then multiplying by the number of files.

Definition

Data capacity

Data capacity is the amount of data that can be stored, or the amount of storage required for a particular set of data.

For example, if one image is 2 MiB and a phone must store 500 images, the storage required is 1000 MiB. You may then convert that into GiB if needed.

Example

Calculating total storage for many files

A security camera stores 96 images each hour. Each image is 750 KiB. Calculate the storage needed for 24 hours in MiB.

Find the number of images stored in 24 hours:
$96 \times 24 = 2304 \text{ images}$
Multiply by the size of one image to get the total in KiB:
$2304 \times 750 = 1,728,000 \text{ KiB}$
Convert KiB to MiB by dividing by 1024:
$\div 1024 = 1687.5 \text{ MiB}$
The storage required is 1687.5 MiB.

How many files will fit?

Sometimes the question gives a storage device capacity and asks how many files can be stored.

In that case:

file\text{number of files} = \text{total capacity} \div \text{size of one file}

If the result is not a whole number, round down, because you cannot store part of a file and count it as a complete stored file.

Example

Finding how many files fit on a device

A memory card has 2 GiB available. Each audio file is 4 MiB. How many complete audio files can be stored?

Convert the total capacity into the same unit as the file size. Since 1 GiB = 1024 MiB:
$\times 1024 = 2048 \text{ MiB}$
Divide the total capacity by the size of one file:
$2048 \div 4 = 512$
Interpret the result as a number of complete files. Since the answer is already a whole number, the card can store 512 audio files.

Common Mistake

Mixing units in one calculation

Do not divide GiB by MiB, or bits by bytes, without converting first. Put both values into the same unit before comparing or dividing.

Exam technique

In the exam

Write the expression before calculating, so the examiner can see which quantities you are using.
Convert units carefully: bits to bytes uses 8, but KiB to MiB and MiB to GiB use 1024.
If asked how many whole files fit, convert to the same unit, divide, then round down if necessary.

Self review

Check yourself

Why is 1 KiB equal to 1024 bytes rather than 1000 bytes?
What is the difference between b and B in storage units?
A file is 8192 bits. How many bytes is it, and how many KiB is it?

Binary multiples and file size calculations

What you'll learn

The storage units Edexcel expects: bit, nibble, byte, KiB, MiB, GiB and TiB.
Why GCSE Computer Science uses binary multiples based on 1024, not decimal thousands.
How to convert between storage units accurately.
How to construct expressions for file size and total data capacity.

Starting point: data is stored as bits

Computers store data using binary, a number system that uses only two digits: 0 and 1. Each 0 or 1 is a bit.

Definition

Bit

A bit is a single binary digit: either 0 or 1. It is the smallest unit of data storage.

Bits are grouped into larger units so that file sizes are easier to describe. It would be awkward to say a small image is 2,457,600 bits every time, so we often convert into bytes, KiB or MiB.

The storage units you need

A nibble is 4 bits. A byte is 8 bits, or 2 nibbles.

After bytes, Edexcel uses binary multiples:

Unit	Symbol	Meaning
bit	b	one binary digit
nibble	—	4 bits
byte	B	8 bits
kibibyte	KiB	1024 bytes
mebibyte	MiB	1024 KiB
gibibyte	GiB	1024 MiB
tebibyte	TiB	1024 GiB

Definition

Binary multiple

A binary multiple is a storage unit based on powers of 2. In this topic, each larger unit from KiB upwards is 1024 times the previous unit because 1024 is 2 to the power of 10.

This ladder shows the direction of the conversions.

Storage unit conversion ladder from bits to tebibytes

Key Idea

The unit ladder

Move towards smaller units by multiplying. Move towards larger units by dividing. From byte upwards, the factor is 1024 each time.

Common Mistake

Using 1000 instead of 1024

For this Edexcel GCSE topic, use KiB, MiB, GiB and TiB, not decimal kB/MB/GB/TB. So 1 KiB is 1024 bytes, not 1000 bytes.

Bits and bytes: the capital letter matters

The symbol b usually means bit. The symbol B means byte.

That tiny difference changes the size by a factor of 8.

64 b means 64 bits.
64 B means 64 bytes, which is 512 bits.

Tip

Sanity check

If a file size is in bits, divide by 8 to get bytes. If it is in bytes, multiply by 8 to get bits.

Example

Converting bits to bytes

A small data item is 2048 bits. Convert this to bytes.

Use the relationship 1 byte = 8 bits, so the number of bytes is found by dividing the number of bits by 8.
Substitute the value:
$bytes=2048÷8\text{bytes} = 2048 \div 8$
Calculate the result:
$bytes=256\text{bytes} = 256$
So 2048 bits is 256 B.

Converting between KiB, MiB, GiB and TiB

From bytes upwards, each step uses 1024.

So:

bytes to KiB: divide by 1024
KiB to MiB: divide by 1024
MiB to GiB: divide by 1024
GiB to TiB: divide by 1024

Going the other way, multiply by 1024 each step.

Example

Converting MiB to bytes and bits

Convert 3 MiB into bytes and then into bits.

Convert MiB to KiB by multiplying by 1024:
$\times 1024 = 3072 \text{ KiB}$
Convert KiB to bytes by multiplying by 1024 again:
$3072 \times 1024 = 3,145,728 \text{ B}$
Convert bytes to bits by multiplying by 8:
$\times 8 = 25,165,824 \text{ bits}$
So 3 MiB is 3,145,728 B, or 25,165,824 bits.

File size: build the expression first

A file size is the amount of storage needed to store a file.

The main GCSE skill is not just “doing sums”; it is choosing the correct expression from the information given.

Key Idea

General file size pattern

A file size calculation usually follows this pattern: number of items multiplied by bits needed for each item. Then convert the answer into the unit requested.

For example:

For text, the items are characters. A character is one symbol such as A, 7, ? or a space.
For a bitmap image, the items are pixels. A pixel is one tiny coloured square in the image. The colour depth is the number of bits used for each pixel.
For sound, the items are samples. A sample is one measurement of the sound wave. The sample rate tells you how many samples are recorded per second, and the bit depth tells you how many bits are used per sample.

You may not need all of these formulae in one question, but the same structure appears again and again.

Common expressions

For text:

character\text{file size in bits} = \text{number of characters} \times \text{bits per character}

For a bitmap image:

depth\text{file size in bits} = \text{width} \times \text{height} \times \text{colour depth}

For sound:

depth\text{file size in bits} = \text{sample rate} \times \text{duration} \times \text{bit depth}

If the question gives multiple channels, such as stereo sound, include that too:

depth×channels\text{file size in bits} = \text{sample rate} \times \text{duration} \times \text{bit depth} \times \text{channels}

Common Mistake

Only include what the question gives you

GCSE file size questions usually ignore extra storage such as file headers or metadata unless the question explicitly tells you to include it.

Example

Calculating a bitmap image file size

An uncompressed bitmap image is 640 pixels wide and 480 pixels high. Each pixel uses 8 bits. Calculate the file size in KiB.

Construct the expression in bits using width, height and colour depth:
$size=640×480×8\text{file size} = 640 \times 480 \times 8$
Calculate the number of bits:
$640 \times 480 \times 8 = 2,457,600 \text{ bits}$
Convert bits to bytes by dividing by 8:
$\div 8 = 307,200 \text{ B}$
Convert bytes to KiB by dividing by 1024:
$\div 1024 = 300 \text{ KiB}$
The file size is 300 KiB.

Data capacity requirements

A data capacity requirement is the total amount of storage needed for a system, device or collection of files.

This often means calculating the size of one file, then multiplying by the number of files.

Definition

Data capacity

Data capacity is the amount of data that can be stored, or the amount of storage required for a particular set of data.

For example, if one image is 2 MiB and a phone must store 500 images, the storage required is 1000 MiB. You may then convert that into GiB if needed.

Example

Calculating total storage for many files

A security camera stores 96 images each hour. Each image is 750 KiB. Calculate the storage needed for 24 hours in MiB.

Find the number of images stored in 24 hours:
$96 \times 24 = 2304 \text{ images}$
Multiply by the size of one image to get the total in KiB:
$2304 \times 750 = 1,728,000 \text{ KiB}$
Convert KiB to MiB by dividing by 1024:
$\div 1024 = 1687.5 \text{ MiB}$
The storage required is 1687.5 MiB.

How many files will fit?

Sometimes the question gives a storage device capacity and asks how many files can be stored.

In that case:

file\text{number of files} = \text{total capacity} \div \text{size of one file}

If the result is not a whole number, round down, because you cannot store part of a file and count it as a complete stored file.

Example

Finding how many files fit on a device

A memory card has 2 GiB available. Each audio file is 4 MiB. How many complete audio files can be stored?

Convert the total capacity into the same unit as the file size. Since 1 GiB = 1024 MiB:
$\times 1024 = 2048 \text{ MiB}$
Divide the total capacity by the size of one file:
$2048 \div 4 = 512$
Interpret the result as a number of complete files. Since the answer is already a whole number, the card can store 512 audio files.

Common Mistake

Mixing units in one calculation

Do not divide GiB by MiB, or bits by bytes, without converting first. Put both values into the same unit before comparing or dividing.

Exam technique

In the exam

Write the expression before calculating, so the examiner can see which quantities you are using.
Convert units carefully: bits to bytes uses 8, but KiB to MiB and MiB to GiB use 1024.
If asked how many whole files fit, convert to the same unit, divide, then round down if necessary.

Self review

Check yourself

Why is 1 KiB equal to 1024 bytes rather than 1000 bytes?
What is the difference between b and B in storage units?
A file is 8192 bits. How many bytes is it, and how many KiB is it?

Binary multiples and file size calculations

What you'll learn

Starting point: data is stored as bits

The storage units you need

Bits and bytes: the capital letter matters

Converting between KiB, MiB, GiB and TiB

File size: build the expression first

Common expressions

Data capacity requirements

How many files will fit?

You've reached the end

Binary multiples and file size calculations

What you'll learn

Starting point: data is stored as bits

The storage units you need

Bits and bytes: the capital letter matters

Converting between KiB, MiB, GiB and TiB

File size: build the expression first

Common expressions

Data capacity requirements

How many files will fit?

You've reached the end

Binary multiples and file size calculations

What you'll learn

Starting point: data is stored as bits

The storage units you need

Bits and bytes: the capital letter matters

Converting between KiB, MiB, GiB and TiB

File size: build the expression first

Common expressions

Data capacity requirements

How many files will fit?

You've reached the end

Binary multiples and file size calculations Revision Guide

Binary multiples and file size calculations

What you'll learn

Starting point: data is stored as bits

The storage units you need

Bits and bytes: the capital letter matters

Converting between KiB, MiB, GiB and TiB

File size: build the expression first

Common expressions

Data capacity requirements

How many files will fit?

You've reached the end

Binary multiples and file size calculations Revision Guide