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Advanced Conversions

There are many way to do conversions from one unit to another (like minutes to hours or feet to miles).

When problems are complicated, we like to use the "you can multiply anything by a fraction that equals 1" rule, to complete conversions.  The process is called dimensional analysis and you many have learned a strategy like this in chemistry. But, it's actually simpler than it sounds.

Before we explain more, let's give an example:

A 4 year old will turn 5 in 6 days.  She wants to know how many minutes 6 days is. Let's multiply some fractions to find the answer.  Start with your base: 6 days.

$\require{cancel}\dfrac{6 \text{ days}}{1}$

Then multiply by a fraction that equations to 1.  There are 24 hours in one day.  Because 24 hours and 1 day are equal, the fraction $\dfrac{24 \text{ hours}}{1 \text{ day}}$ equals 1 (so does the fraction $\dfrac{1 \text{ day}}{24\text{hours}}$).  In this case, you want to use the fraction with the unit "day" (which you want to get rid of) on the bottom where it will cross-cancel (because the "day" in your original fraction is in the numerator). 

 $\dfrac{6 \text{ days}}{1}\times \dfrac{24 \text{ hours}}{1 \text{ day}}$

Now we want to get rid of the unit "hours." So, we multiply by $\dfrac{1 \text{ hour}}{60 \text{ minutes}}$, which equals 1.  But, because this time we want to eliminate the unit "hours" and it's on top, we actually want to multiply by the reciprocal (which also equals 1): $\dfrac{60 \text{ minutes}}{1 \text{ hour}}$

$\dfrac{6 \text{ days}}{1}\times \dfrac{24 \text{ hours}}{1 \text{ day}} \times \dfrac{60 \text{ minutes}}{1 \text{ hour}}$

Now, we do the multiplication problem.  When we cross-cancel units, we will be left with just one unit label, the one we want: minutes.

$\dfrac{6 \cancel{\text{ days}}}{1}\times \dfrac{24 \cancel{\text{ hours}}}{1 \cancel{\text{ day}}} \times \dfrac{60 \boxed{\text{ minutes}}}{1 \cancel{\text{ hour}}}=\dfrac{8640 \text{ minutes}}{1}=8640 \text{ minutes}$

This process works for many types of conversions.  You just have to remember: when you multiply by a fraction make sure that it equals 1 (the top and bottom are equivalent) and put units on opposite sides of the fractions to were they already are in the problem, so that they cross-cancel.

Practice Problems:

  • Advanced Conversions

    1. How many minutes are in 1 week?
    2. How many hours are there in 3 years?
    3. How many inches are there in 12 yards?
    4. How many yards are there in 5 inches?
    5. How many hours are in January?
    6. If there are 12 crayons in a box, and 60 crayon boxes in a carton, how many crayons are in 5 crayon cartons?
    7. If a teaspoon is $\dfrac{1}{3}$ of a tablespoon and there are 16 tablespoons in a cup, how many teaspoons are in a quarter of a cup?
    8. In an imaginary country, 3 daks = 6 goops.  2 goops=9 dings.  If you have 12 dings, how many daks do you have?
    9. A company sells candy in 2 decameter sticks. A restaurant wants to cut those sticks into 1 centimeter bites. If there are 10 meters in a decameter and 100 centimeters in a meter, how many "bites" does a teacher get out of 4 sticks?
    10. Fabric comes in 50 yard bolts. A seamstress makes doll skirts that use 9 inches of fabric each.  How many doll skirts can the seamstress make from one bolt?

    Answer Key: