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# Units and unit conversion on the SAT test

## Studying units and unit conversion

On the SAT test units and unit conversion topic is part of problem solving and data analysis subscore that includes 9 advanced topics (see the full topics list on the top menu).

Units and unit conversion topic is the third topic of problem solving and data analysis subscore. It is recommended to start learning problem solving and data analysis subscore with its first topic called ratios, rates and proportions.

Units and unit conversion topic is divided into sections from easy to difficult (the list of the sections appears on the left menu). Each section includes detailed explanations of the required material with examples followed by a variety of self-practice questions with solutions.

Finish studying heart of algebra subscore topics before you study this topic or any other problem solving and data analysis subscore topic. (Heart of algebra subscore includes basic algebra topics which knowledge is required for understanding problem solving and data analysis subscore topics).

### Units and unit conversion -summary

A unit is used to measure a quantity, while different kinds of quantities measured in different units.

Unit conversion changes a given quantity to a different unit of measurement, so that the relative amount does not change. Unit conversion is calculated with unit equivalencies.

Unit equivalencies show how the units of the same kind of quantity relate to each other.
Time Unit equivalencies measure time intervals that express duration (for example: 1 minute= 60 seconds).
Distance Unit equivalencies measure length (for example: 1 kilometer=1,000 meters ).
Mass Unit equivalencies measure weight of objects (for example: 1 kilogram=1,000 grams).
Volume Unit equivalencies measure capacity of an object or space in three dimensions (for example: 1 liter=1,000 milliliters).

Unit conversion is calculated by multiplication of the original quantity by a conversion factor of 1 and simplifying so that we are left with the desired units without the original units.

To Convert between 2 units a and c using a third unit b, we need to multiply by 2 conversion factors: a conversion factor between unit a and unit b and a conversion factor between unit b and unit c.

To Convert units that appear within rates, we need to construct one or two conversion factors to cancel the units that are given and not desired and create the units that are not given and desired.

## Unit conversion and unit equivalency definitions

A unit is used to measure a quantity, while different kinds of quantities measured in different units. For example: distance is measured in miles, weight is measured in kilograms and time is measured in hours. There can be many unites for measuring the same kind of quantity. For example: time can be measured in hours, minutes, or seconds.

Unit conversion changes a given quantity to a different unit of measurement, so that the relative amount does not change. For example: 0.5 hours can be converted to 30 minutes (the relative amount stayed the same). Unit conversion is calculated by multiplication of the original quantity by a conversion factor of 1 and simplifying so that we are left with the desired units without the original units.

Unit equivalencies show how the units of the same kind of quantity relate to each other. For example: 1 hour is equal to 60 minutes. To convert a given quantity to a desired unit, you first need to know the unit equivalency between the given unit and the desired unit.

### Unit equivalencies lists

Below are lists of unit equivalences for time, distance, mass and volume.

Time Unit equivalencies:
Units of time measure time intervals that express duration.

1 minute= 60 seconds
1 hour= 60 minutes
1 day= 24 hours
1 week= 7 days
1 year= 12 months
1 year= 52 weeks
1 year= 365 days

Distance Unit equivalencies:
Units of distance measure length.

1 centimeter (cm)=10 millimeters (mm)
1 inch (in)= 2.54 centimeters (cm)
1 foot (ft)= 12 inches (in)
1 yard (yd)= 3 feet (ft)
1 meter (m)= 100 centimeters (cm)
1 kilometer (km)= 1,000 meters (m)
1 mile (mi)= 1.61 kilometers (km)

Mass Unit equivalencies:
Units of mass measure weight of objects.

1 gram (g)= 1,000 milligrams (mg)
1 pound = 453.59 grams (g)
1 kilogram (kg)= 2.2 pounds
1 kilogram (kg)= 1,000 grams (g)
1 ton (t)= 1,000 kilograms (kg)

Volume Unit equivalencies:
Units of volume measure capacity of an object or space in three dimensions, the objects can be fluids (like water) or bulk goods (like flour). Since volume is measured in three dimensions, the units of measure for volume are cubic units.

1 liter (l)= 1,000 milliliters
1 kiloliter= 1,000 liters

### Calculating unit conversion

The unit conversion changes a given quantity to a different unit of measurement, so that the relative amount does not change.

The goal of the unit conversion is to transform the given quantity to desired units of measure, therefore we need to cancel the given units and create the desired units:
The given quantity has the given units in the nominator. To cancel the given units in the numerator we need to divide the given quantity by the given units.
The new quantity should have the desired units in the nominator. To create the desired units in the numerator we need to multiply the given quantity by the desired units.
Since the relative amount must stay the same, the only operation we can do is to multiply by 1 using unit equivalencies.

Therefore, unit conversion is calculated by multiplication of the original quantity by a conversion factor of 1 and simplifying so that we are left with the desired units without the original units.

#### Unit conversion steps

Remember to write the name of the unit near each value!

Step 1: Write the unit equivalence between the given units and the desired units.

Step 2: Write the conversion factor of 1 so that the desired units are in the numerator and the given unit are in the denominator.

Step 3: Write the conversion expression that is equal to the given quantity multiplied by the conversion factor of 1.

Step 4: Simplify the expression from step 3 so that you cancel out the given units and are left with the desired units.

Consider the following example:

Convert 2.7 meters (m) to centimeters (cm).

Step 1: Writing the unit equivalence between meters and centimeters:
We need to convert 2.7 meters, therefore we should write 1 meter in centimeters: 1 meter= x centimeters.
We know that 1 meter= 100 centimeters

Step 2: Writing the conversion factor of 1 so that the centimeters are in the numerator meters are in the denominator:
100 centimeters / 1 meter= 1

Step 3: Writing the conversion expression that is equal to the given quantity multiplied by the conversion factor of 1:
2.7 meters = 2.7 meters * 100 centimeters / 1 meter

Step 4: Simplifying the expression from step 3 canceling meters and leaving centimeters:
2.7 meters * 100 centimeters / 1 meter= 2.7*100 centimeters = 270 centimeters.

### Converting between 2 units using a third unit

In these questions we are asked to convert unit a to unit c, while we are given 2 unit equivalencies: the unit equivalency between unit a and unit b and the unit equivalency between unit b and unit c. Therefore, to convert from unit a to unit c we must include their unit equivalency with unit b.

To Convert between 2 units using a third unit, we need to solve using the steps introduced above. In these questions we need to multiply by 2 conversion factors: a conversion factor between unit a and unit b and a conversion factor between unit b and unit c.

We are given unit a in the numerator and we need to be left with unit c in the numerator. To do so we must divide by unit a and multiply by unit c, therefore we need unit c in the numerator and unit a in the denominator. Since we are not given the direct unit equivalency between unit a and unit c, we must include their unit equivalencies with unit b. We know that unit b must cancel out, therefore we write unit b in the numerator and the denominator:

unit c                 unit b
_______    *   ________
Unit b                unit a

Consider the following example:

1 kilometer (km)= 1,000 meters (m)
1 mile (mi)= 1.61 kilometers (km)

How many meters are in 1 mile?

Step 1: Writing the unit equivalence between miles and meters:
We need to convert miles to meters, but we are given the conversion factor between miles and kilometers and the conversion factor between kilometers and meters.
1 kilometer (km)= 1,000 meters (m)
1 mile (mi)= 1.61 kilometers (km)

Step 2: Writing the 2 conversion factors of 1:
We know that we must divide by miles and multiply be meters (so that miles unit will cancel out and meters unit will be created).

meters
_______  * _______
miles

We don’t need kilometers in the answer and we don’t have them in the given quantity, but we must use them in the conversion factors. Therefore, we must create kilometers and then cancel them out, to do so we need to multiply and divide by kilometers.

kilometers           meters
___________    *     _________
miles                     kilometers

Now we can put values from the given unit equivalencies into the conversion factors:
1 kilometer (km)= 1,000 meters (m)
1 mile (mi)= 1.61 kilometers (km)

1.61 kilometers          1,000 meters
________________   *     _____________
1 miles                         1 kilometers

Step 3: Writing the conversion expression that is equal to the given quantity multiplied by 2 conversion factors of 1:

1 mile= 1 mile     1.61 kilometers          1,000 meters     = 1.61 * 1,000 = 1,610 meters
*  _______________    *    ______________
1 mile                      1 kilometers

### Converting units that appear within rates

Each rate is composed from 2 units, one in the numerator and the other in the denominator. Remember to write the name of the units near the values in the numerator and the denominator of the rate.

To Convert units that appear within rates, we need to solve using the steps introduced above. In these questions we need to construct one or two conversion factors to cancel the units that are given and not desired and create the units that are not given and desired.

Consider the following example:

Convert 1 dollars per pound to cents per kilograms.

Step 1: Writing the unit equivalence between the units:
1 kilogram (kg)= 2.2 pounds
1 dollar= 100 cents

Step 2: Writing the conversion factors of 1:

In this question we need to convert between 2 rates and each rate is composed from different units. Therefore, we need to cancel 2 units that appear in numerator and the denominator of the given rate (dollars per pounds) and create 2 units that appear in the numerator and the denominator of the desired rate (cents per kilograms).

pounds             cents
________   *   ___________
dollars            kilograms

We know the unit equivalence between dollars and cents and between kilograms and pounds:

2.2 pounds          100   cents
____________   *   ____________
1 kilogram               1 dollar

Step 3: Writing the conversion expression that is equal to the given quantity multiplied by the conversion factors of 1:

1 dollar           1 dollar           2.2 pound       100 cents
________  =   __________  *    ___________  *  __________
1 pound         1 pound           1 kilogram        1 dollar

Step 4: Simplifying the expression from step 3 canceling dollars and pounds and leaving cents and kilograms:

1 dollar           2.2 pounds          100 cents         2.2*100 cents          220 cents
_________  *   _____________  *   ___________  =  _______________  =   ____________
1 pound           1 kilogram            1 dollar              1 kilogram              1 kilogram

You just finished studying units and unit conversion topic, the third topic of problem solving and data analysis subscore!

Continue studying the next problem solving and data analysis subscore topic- linear and exponential growth .