[fs-toc-h2]Single Unit conversions
Some questions on the SAT will ask you to convert from one unit to another, the most prevalent type of question asks you to convert a singular unit that measures the same thing, such as pounds to kilograms, inches to feet, or feet to meters.
The easiest way to solve most of these involves creating a proportion. Let's say Johnny just harvested 55 kilograms of apples, but needs to know how much that weighs in pounds, where 1 kilogram = 2.2 pounds.
NOTE: The SAT will always give you a conversion factor, you are not expected to memorize them.
We know that the ratio between the weight in pounds to the weight in kilograms is equal to 2.2 kilograms to 1 pounds.
\[ \frac{x \text { lbs}}{55 \text { kg}} = \frac{1 \text { lb}}{2.2 \text {1 kg}}\]
Isolating x results in
\[x \text{ lbs} = 55 \cancel{\text{ kg}} \cdot \frac{2.2 \text{ lbs}}{1 \cancel{\text{ kg}}}\]
Notice, if done correctly, all the units should cancel out other than the unit you’re trying to solve for. In this case the kg unit is canceled and all we’re left with is pounds.
Knowing this, there's a quicker way you could do this known as dimensional analysis (commonly used in chemistry classes). While it may not seem much quicker now, when you need to introduce more units, it practically becomes essential.
First step is to write what your wanting to find and what you’re given on opposite sides of the equal sign
\[x \text{ lbs} = 55 \text{ kg}\]
Since we already know the ratio, but the unit that your solving for (lbs) on top so the unit you’re given (kg) cancels out, the easiest way to remember this is wanthave.
\[x \text{ lbs} = 55 \text{ kg} \cdot \frac{2.2 \text{ lbs}}{1 \text{ kg}}\]
When plugged into a calculator, this comes out to be approximately 121 pounds.
[fs-toc-h4]Example 1
First step is to find the weight of each slice in pounds. The pizza is first sliced in half
\[\frac{3 \text{ lbs}}{2} = 1.5 \text{ lb}\]
Then that half is sliced into thirds
\(\frac{1.5 \text{ lbs}}{3} = 0.5 \text{ lbs}\), meaning each slice weight \(0.5 \text{ lbs}\)
We can now set up our conversion. Again, first write out what you want and what you have
\[x \text{ oz} = 0.5 \text{ lbs}\]
Now put the unit you want on top and the unit you have on the bottom of the ration (want/have)
\[x \text{ oz} = 0.5 \cancel{\text{ lbs}} \cdot \frac{16 \text{ oz}}{1 \cancel{\text{ lbs}}}\]
Plug this into your calculator to get 8 ounces per slice, making choice C the correct answer.
For more on single unit conversions, and an introduction to multi-step conversions, watch the video below.
[fs-toc-h2]Multi-step Conversions (Dimensional Analysis)
In the previous example we only had to convert from one unit to another. Though, there are cases where you’ll have to do multiple conversions on one number
Let's say Johnny traveled 0.5 kilometers to get to the apple orchard, how far would he have traveled in feet?
We would most probably not be given a direct conversion rate from kilometers to inches, but we can convert from kilometers to meters, meters to feet, then feet to inches. This really is no different that a single conversion as we did before, all we’re doing is adding multiple conversion rates to the same problem. Again, start out with what you want and what you have first
\[x \text{ inches} = 0.5 \text{ kilometers}\]
The first conversion will be from kilometers to meters (1 km = 1,000 m)
\[x \text{ inches} = 0.5 \cancel{\text{ km}} \cdot \frac{1{,}000 \text{ m}}{1 \cancel{\text{ km}}}\]
While you could immediately compute this here, then repeat the conversion for three other steps, it can be done quicker by making a singular equation. We know that after meters, we have to feet ( 1 m = 3.3 ft ), meaning we can simply add on that conversion to the equation.
Since kilometers has canceled out, and the only unit that's left is meters, we have to put meters at the bottom and feet at the top (remember, want/have)
\[x \text{ inches} = 0.5 \cancel{\text{ km}} \cdot \frac{1{,}000 \cancel{\text{ m}}}{1 \cancel{\text{ km}}} \cdot \frac{3.3 \text{ ft}}{1 \cancel{\text{ m}}}\]
Computing this out will get us the distance in feet, but we have one more step, feet to inches (1 ft = 12 in)
Again, we’re left with feet, we want to convert to inches, so feet is on the bottom, inches are on top.
\[x \text{ inches} = 0.5 \cancel{\text{ km}} \cdot \frac{1{,}000 \cancel{\text{ m}}}{1 \cancel{\text{ km}}} \cdot \frac{3.3 \cancel{\text{ ft}}}{1 \cancel{\text{ m}}} \cdot \frac{12 \text{ in}}{1 \cancel{\text{ ft}}}\]
After all units cancel out, we’re left with inches, the desired unit. Computing this in a calculator will give you approximately 19,800 inches.
[fs-toc-h4]Example 1
Lets write out what we want and what we have first.
\[x \text{ feet} = 75 \text{ roman digits}\]
Since we’re only given the conversion rate including digits is digitals to pes, thats what we have to include first. We have digits and want pes
\[x \text{ feet} = 75 \text{ roman digits} \cdot \frac{1 \text{ pes}}{16 \text{ digits}}\]
Now we are able to convert from pes to inches (1 pes = 11.65 in), then inches to feet (12 in = 1ft)
\[x \text{ feet} = 75 \cancel{\text{ roman digits}} \cdot \frac{1 \cancel{\text{ pes}}}{16 \cancel{\text{ digits}}} \cdot \frac{11.65 \cancel{\text{ in}}}{1 \cancel{\text{ pes}}} \cdot \frac{1 \text{ ft}}{12 \cancel{\text{ in}}}\]
\[x \text{ feet} = 4.55\]
The can also be asked in different ways though, look at the example below.
IMPORTANT NOTE: and important equation you may need to know is the distance/speed/time equation: \(d = st\)
If i'm given a speed of 10 m/s, and a time of 50 seconds, the total distance traveled would be
\[d = 10 \cdot 50\]
\[d = 500 \text{ meters}\]
[fs-toc-h4]Example 2
There are two ways to think about this, using either the distance equation just given, or treating it like a normal unit conversion. No matter how you approach it though, you’ll actually end up with the same equation. Firstly, let's treat this like a multi-step unit conversion, by writing out what I want and what I have
\[x \text{ meters} = 4 \text{ minutes}\]
Be careful here, you may be eager to use the speed ratio given to us here, but realize that our time is in minutes and the speed is in meters per second. We have to convert 4 minutes into seconds first.
\[x \text{ meters} = 4 \text{ minutes} \cdot \frac{60 \text{ seconds}}{1 \text{ minute}}\]
While seconds and meters obviously don't measure the same thing, as long as you can get your units to cancel you’ll end up with the correct answer. Above, we have seconds and we want meters, meaning our ratio will have meters on top and seconds on the bottom.
\[x \text{ meters} = 4 \cancel{\text{ minutes}} \cdot \frac{60 \cancel{\text{ seconds}}}{1 \cancel{\text{ minute}}} \cdot \frac{25 \text{ meters}}{13.7 \cancel{\text{ seconds}}}\]
This comes out to approximately 437, which is closest to answer choice B
Now if you notice, our final equation is actually just the distance/speed/time equation
\[x \text{ meters (distance)} = 4 \text{ minutes} \cdot \frac{60 \text{ sec}}{1 \text{ min}} \text{ (time)} \cdot \frac{25 \text{ meters}}{13.7 \text{ sec}} \text{ (speed)}\]
No matter how you approach this, you should get the same answer.
[fs-toc-h2]Speed Conversions
We did dabble in a bit of these above, but some questions over speed can be a bit different. While things can get a bit confusing here, if you keep track of your units, you will be fine.
Let's say Johnny ran to the farmers market, he traveled 84 meters in 41 seconds. What was his speed in miles per hour?
First step is to right out what we have and want
\[x \frac{\text{miles}}{\text{hour}} = \frac{84 \text{ meters}}{41 \text{ seconds}}\]
When you have two different units that have to be converted, take it one at a time. You can convert from miles to meters without worrying about the time at all, then convert from seconds to hours without having to worry about the distance.
Lets take care of our meters to miles conversion rates first (1 mile = 1609 meters)
\[x \, \frac{\text{miles}}{\text{hour}} = \frac{84 \cancel{\text{ meters}}}{41 \text{ seconds}} \cdot \frac{1 \text{ mile}}{1{,}609 \cancel{\text{ meters}}}\]
Now that we’ve taken care of distance, our second conversion will take care of time. Though, instead of putting what we want on top and we have on the bottom, it has to be switched. Since seconds is originally on the bottom of our fraction, it has to be on top in the conversion ratio in order for the seconds unit to cancel out. (3600 sec = 1 hr)
\[ x \, \frac{\text{miles}}{\text{hour}} = \frac{84 \cancel{\text{ meters}}}{41 \cancel{\text{ seconds}}} \cdot \frac{1 \text{ mile}}{1{,}609 \cancel{\text{ meters}}} \cdot \frac{3{,}600 \cancel{\text{ seconds}}}{1 \text{ hour}}\]
Now the only units we’re left with after canceling everything out is miles/hours, meaning we’ve written our equation right. Only step left is plugging it all into a calculator, giving us an answer of \(4.58 \, \frac{\text{miles}}{\text{hour}}\)
We highly recommend watching the video below, as it covers many examples and different units.
[fs-toc-h4]Example 1
Taking a look at this, you may notice this is actually less steps than our previous example. Ive said it like a hundred times this lesson, but let's write out what we have and want.
\[x \, \frac{\text{miles}}{\text{hour}} = \frac{11 \text{ miles}}{26 \text{ minutes}}\]
Since our distance units are the exact same, we only have to worry about the time, which is going from minutes to hours
\[x \, \frac{\text{miles}}{\text{hour}} = \frac{11 \text{ miles}}{26 \cancel{\text{ minutes}}} \cdot \frac{60 \cancel{\text{ minutes}}}{1 \text{ hour}}\]
Remember, we put minutes on top so the unit can cancel. After calculating, we get the final answer of \(25.4 \, \frac{\text{miles}}{\text{hour}}\)
[fs-toc-h4]Example 2
Sometimes you’ll be faced with units you aren’t familiar with, i mean posters per hour isn't something you run into everyday. Though, the process remains the same: write out all your units and make sure everything cancels out
\[x \, \frac{\text{posters}}{\text{hour}} = \frac{42 \text{ posters}}{\text{minute}}\]
We have to cancel out the minutes unit and convert it to hours, so minutes will be on top and hours will be on the bottom
\[x \, \frac{\text{posters}}{\text{hour}} = \frac{42\text{ posters}}{\cancel{\text{minute}}} \cdot \frac{60 \cancel{\text{ minutes}}}{1 \text{ hour}}\]
Which comes out to 2520 posters per hour
[fs-toc-h2]Area Conversions (square units)
If you want to convert one squared unit to another in an area problem (such as feet squared to inches squared), you can't simply use the given conversion rate as it.
For example, if a garden has an area 144 square feet, what would the area be in square inches (1 foot = 12 inches)
You can start this out like any other unit conversion
\[x \, \text{in}^2 = 144 \, \text{ft}^2 \cdot \frac{12 \text{ in}}{1 \text{ ft}}\]
Now if you stop here and compute, you will get the wrong answer. The issue here is that your trying to convert square feet by just using the feet conversion rate. To solve this you have to square your entire conversion rate and then solve.
\[x \, \text{in}^2 = 144 \, \text{ft}^2 \cdot \left(\frac{12 \text{ in}}{1 \text{ ft}}\right)^2\]
\[x \, \text{in}^2 = 144 \, \cancel{\text{ft}}^2 \cdot \frac{144 \, \text{in}^2}{1 \, \cancel{\text{ft}}^2}\]
Now the feet squared unit cancels out, and all your left with is inches squared. Compute to get an answer of \(20{,}736 \, \text{in}^2\)
[fs-toc-h4]Example 1
First, write out what you want, need, and your conversion rate
\[x \, \text{yd}^2 = 4.36 \, \text{mi}^2 \cdot \frac{1{,}760 \, \text{yd}}{1 \, \text{mi}}\]
Now square your conversion rate and distribute the exponent
\[x \, \text{yd}^2 = 4.36 \, \text{mi}^2 \cdot \left(\frac{1{,}760 \, \text{yd}}{1 \, \text{mi}}\right)^2\]
\[x \, \text{yd}^2 = 4.36 \, \text{mi}^2 \cdot \frac{3{,}097{,}600 \, \text{yd}^2}{1 \, \text{mi}^2}\]
Finally compute to get an answer of \(13{,}505{,}536 \, \text{yd}^2\), making choice D the correct answer.