﻿ Multi-Step Conversions
MULTI-STEP CONVERSIONS
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
• PRACTICE (online exercises and printable worksheets)
Want more details, more exercises?
• For simpler conversion problems, try One-Step Conversions.

Any unit of length can be converted to any other unit of length.
Any unit of time can be converted to any other unit of time.
Any unit of volume can be converted to any other unit of volume.
Any unit of mass or weight can be converted to any other unit of mass or weight.

This lesson covers multi-step conversions: that is, you will probably need to multiply by 1 more than once.
For simpler problems and a discussion of the concepts involved, see the prior section, One-Step Conversions.

The examples below might be all you need.
If you'd like more guidance, see the Suggestions for Writing Down Unit Conversion Problems further down this page.

There may be more than one correct way to reach your final answer.
Since the conversion factors between metric units and English units are approximate,
different correct approaches may yield slightly different answers, as shown in the example below:

EXAMPLE:
Question: Convert $\,13\,$ yards to centimeters.

$\displaystyle \cssId{s24}{13\text{ yd}} \cssId{s25}{= 13\text{ yd}} \cssId{s26}{\cdot\frac{3\text{ ft}}{1\text{ yd}}} \cssId{s27}{\cdot\frac{12\text{ in}}{1\text{ ft}}} \cssId{s28}{\cdot\frac{2.5\text{ cm}}{1\text{ in}}} \cssId{s29}{= 1170\text{ cm}}$

An equally correct approach that yields a slightly different answer is:

$\displaystyle \cssId{s31}{13\text{ yd}} \cssId{s32}{= 13\text{ yd}} \cssId{s33}{\cdot\frac{1\text{ m}}{1.1\text{ yd}}} \cssId{s34}{\cdot\frac{100\text{ cm}}{1\text{ m}}} \cssId{s35}{= 1181.8\text{ cm}}$

If more accurate conversion factors were used, then these results would agree more closely.

Feel free to use scrap paper and a calculator to compute your answers.
As shown in the example above,
depending on how you do the unit conversion,
you may get a slightly different answer than the answer reported here.

Use only the conversion information given in the Unit Conversion Tables to compute your answers.

All answers are either exact, or rounded to six decimal places.
It is possible to get $\,0.000000\,$ as an answer.

EXAMPLES:
Question:
Convert $\,3\text{ cm}\,$ to $\,\text{ mm}$.

Possible Solution Process:
$\displaystyle \cssId{s52}{3\text{ cm}} \cssId{s53}{= 3\cancel{\text{ cm}}} \cssId{s54}{\cdot\frac{1\bcancel{\text{ m}}}{100\cancel{\text{cm}}}} \cssId{s55}{\cdot\frac{1000\text{ mm}}{1\bcancel{\text{ m}}}} \cssId{s56}{= \frac{3 \cdot 1000}{100}\text{ mm}} \cssId{s57}{= 30\text{ mm}}$

Answer: $30$

Question:
Convert $\,11\text{ mm}\,$ to $\,\text{ in}$.

Possible Solution Process:
$\displaystyle \cssId{s63}{11\text{ mm}} \cssId{s64}{= 11\cancel{\text{ mm}}} \cssId{s65}{\cdot\frac{1\bcancel{\text{ m}}}{1000\cancel{\text{ mm}}}} \cssId{s66}{\cdot\frac{100\cancel{\text{ cm}}}{1\bcancel{\text{ m}}}} \cssId{s67}{\cdot\frac{1\text{ in}}{2.5\cancel{\text{ cm}}}} \cssId{s68}{= \frac{11 \cdot 100}{1000\cdot 2.5}\text{ in}} \cssId{s69}{= 0.44\text{ in}}$

Answer: $0.44$

## Some People Can Just Do It!

Some people can consistently get correct answers (even in these more difficult unit conversion problems) by just sort of ‘doing it’—multiplying by this, dividing by that—without thinking about things like ‘multiplying by one’ and cancellation of units.

That's wonderful if you can do this! Be aware, however, that most people can't. When the accuracy of a result is important, then you may need to justify your answer to a co-worker, teacher, or boss. That is, you may be asked to write work down that convinces them that the answer you've arrived at is correct.

Or, you might come up against a unit conversion problem that is so weird and/or unfamiliar that you just can't figure it out ‘in your head’.

Or, you might be involved in implementing unit conversion features in future computer languages or applications (ones that don't yet exist). You'll need an algorithmic approach that always works.

Therefore, in this author's humble opinion, it is valuable for everyone to master the skills in this lesson—even if you personally don't need to use this systematic approach most of the time.

## Suggestions for Writing Down Unit Conversion Problems (and a fun result)

Way back in the 1970s,  floppy disks were commonly used to store digital data.
A typical floppy disk capacity was 720 kilobytes.

As I write these words (in the year 2017), disk storage is often measured in units of terabytes.

Here is a progression of common digital storage units over the years:

\begin{align} 1 \text{ kilobyte (kB)} &= 1000 \text{ bytes}\cr 1 \text{ megabyte (MB)} &= 1000 \text{ kilobytes}\cr 1 \text{ gigabyte (GB)} &= 1000 \text{ megabytes}\cr 1 \text{ terabyte (TB)} &= 1000 \text{ gigabytes} \end{align}

So, here's a fun question (below).
Its solution is used to suggest some thought processes when writing down unit conversion problems.

How many floppy disks (with 720 kB capacity) would it take to store 1 terabyte of data?
For convenience, we'll use a custom unit, the fdc, to stand for a 720 kilobyte floppy disk capacity.
That is, $\,1 \text{ fdc} = 720 \text{ kB}\,$.

## Writing Down a Unit Conversion Problem

STEPS IN THE UNIT CONVERSION PROBLEM APPLYING EACH STEP TO THE FLOPPY DISK QUESTION
UNIT CONVERSION: FROM WHAT TO WHAT?

What are you starting with?
Call this the ‘start expression’.

What are you ending with?
Call this the ‘end expression’.

Check that the start and end expressions have compatible units.
That is: can they both be used to measure the same type of thing?

For example: are they both units of length? Volume? Time? Weight/mass?
We want to know:
One terabyte equals how many fdc?

Start expression: $\color{green}{1\text{ TB}}$
End expression: (some number of) fdc

Both TB and fdc are used to measure digital storage.
Indeed, both are measured in terms of bytes.
So, they're compatible units.
BEGIN THE UNIT CONVERSION SENTENCE

The unit conversion (going from start expression to end expression) will consist of a single complete mathematical sentence.

Start with a totally obvious statement: $$\cssId{s134}{\color{green}{\text{start expression}}} \cssId{s135}{= \color{green}{\text{start expression}}}$$ Yep—the start expression is equal to itself!
Don't forget to include the units of the start expression on both sides of the equation.
$\color{green}{1\text{ TB}} = \color{green}{1\text{ TB}}$
GET A ‘CHAIN’ THAT TAKES YOU FROM THE START UNIT TO THE END UNIT

You may or may not immediately know the conversion information between the start and end units.

If not, get a ‘chain’ that takes you from start to end, with this requirement: you must know the unit conversion information between consecutive units in the chain.

For example, to go from seconds to days, you might use this chain:

seconds, minutes, hours, days

Why?
We easily know the relationship between seconds and minutes.
We easily know the relationship between minutes and hours.
We easily know the relationship between hours and days.

For some problems, there are different chains that will work.

You may not need to explicitly write down the chain.
Some people can do this part in their head.
We don't immediately know the relationship between TB and fdc (that's the problem).

We can use this chain:

TB, GB, MB, kB, fdc
CONTINUE THE UNIT CONVERSION SENTENCE:
USE THE CHAIN TO MULTIPLY THE START EXPRESSION BY ‘$\,1\,$’ AS MANY TIMES AS NEEDED

Your chain always starts with the unit of start expression.
Your chain always ends with the unit of end expression.
Thus, your chain always has length of two or more.

Let $\,n\,$ denote the length of the chain, so $\,n\ge 2\,$.

Each consecutive pair in the chain produces an appropriate unit conversion fraction.

For example, what fraction does the consecutive pair ‘seconds, minutes’ give?
We know: $\,60 {\text{ sec}} = 1 {\text{ min}}\,$
To convert from seconds to minutes, use the unit conversion fraction:
$$\frac{1 \text{ min}}{60 \text{ sec}} = 1$$
Memory device: whatever you're going TO is on TOP.
Since we're going TO minutes, minutes is on TOP.

Remember: in order to get the number $\,1\,$, the expression on top must equal the expression on the bottom!

With a chain of length $\,n\,$, you'll have $\,n-1\,$ fractions.

Be sure that all the units (except the unit of the end expression) cancel out!

It's a good idea to show the unit cancellations with slashes.
This author often alternates forward slashes ($\,/\,$) and back slashes ($\,\backslash\,$) to make it easier to keep track of things.
$$\cssId{s181}{\color{green}{1\text{ TB}}} \cssId{s182}{= {\color{green}{1}\cancel{\color{green}{\text{ TB}}}}} \cssId{s183}{\cdot\overbrace{\frac{1000\bcancel{\text{ GB}}}{1\cancel{\text{ TB}}}}^{\text{from TB to GB}}} \cssId{s184}{\cdot\overbrace{\frac{1000\cancel{\text{ MB}}}{1\bcancel{\text{ GB}}}}^{\text{from GB to MB}}} \cssId{s185}{\cdot\overbrace{\frac{1000\bcancel{\text{ kB}}}{1\cancel{\text{ MB}}}}^{\text{from MB to kB}}} \cssId{s186}{\cdot\overbrace{\frac{1\text{ fdc}}{720\bcancel{\text{ kB}}}}^{\text{from kB to fdc}}}$$
FINISH THE UNIT CONVERSION SENTENCE:
MULTIPLY THE NUMBERS;

Gather together the numbers and do the arithmetic.

It's a nice touch to line up the equal (or approximately equal to) signs.

Pat yourself on the back for a job well done!
(It's a really good stretching exercise!)
\begin{align} \cssId{s194}{\color{green}{1\text{ TB}}}\ &\cssId{s195}{= {\color{green}{1}\cancel{\color{green}{\text{ TB}}}}} \cssId{s196}{\cdot\overbrace{\frac{1000\bcancel{\text{ GB}}}{1\cancel{\text{ TB}}}}^{\text{from TB to GB}}} \cssId{s197}{\cdot\overbrace{\frac{1000\cancel{\text{ MB}}}{1\bcancel{\text{ GB}}}}^{\text{from GB to MB}}} \cssId{s198}{\cdot\overbrace{\frac{1000\bcancel{\text{ kB}}}{1\cancel{\text{ MB}}}}^{\text{from MB to kB}}} \cssId{s199}{\cdot\overbrace{\frac{1\text{ fdc}}{720\bcancel{\text{ kB}}}}^{\text{from kB to fdc}}}\cr\cr &\cssId{s200}{= \frac{1000\cdot 1000\cdot 1000}{720} \text{ fdc}}\cr\cr &\cssId{s201}{\approx \color{red}{1{,}388{,}889 \text{ fdc}}} \end{align} It takes almost 1.4 million floppy disks (each with 720 kB capacity) to hold a terabyte of data!
Master the ideas from this section

When you're done practicing, move on to:
Rate Problems

CONCEPT QUESTIONS EXERCISE: