What if you had a jar of pennies? You take out a handful that is equal to
the total. Your friend Zoey then takes a handful that is equal to
the amount that remains. What fraction of the total does Zoey take? After completing this Concept, you'll be able to solve real-world problems like this one.
CK-12 Foundation: 0207S Real-World Multiplication and Division (H264)
Using the skills learned in the last two concepts, you now ready to solve real-world problems.
In the chemistry lab there is a bottle with two liters of a 15% solution of hydrogen peroxide
. John removes one-fifth of what is in the bottle, and puts it in a beaker. He measures the amount of
and adds twice that amount of water to the beaker. Calculate the following measurements.
a) The amount of
left in the bottle.
b) The amount of diluted
in the beaker.
c) The concentration of the
in the beaker.
a) To determine the amount of
left in the bottle, we first determine the amount that was removed. That amount was
of the amount in the bottle (2 liters).
of 2 is
The amount remaining is
liter (or 1.6 liters).
There are 1.6 liters left in the bottle.
b) We determined that the amount of the 15%
solution removed was
liter. The amount of water added was twice this amount, or
liter. So the total amount of solution in the beaker is now
liter, or 1.2 liters.
There are 1.2 liters of diluted
in the beaker.
c) The new concentration of
can be calculated.
John started with
liter of 15%
solution, so the amount of
is 15% of
After he adds the water, there is 1.2 liters of solution in the beaker, so the concentration of
or 0.05. To convert to a percent we multiply this number by 100, so the beaker’s contents are 5%
Anne has a bar of chocolate and she offers Bill a piece. Bill quickly breaks off
of the bar and eats it. Another friend, Cindy, takes
of what was left. Anne splits the remaining candy bar into two equal pieces which she shares with a third friend, Dora. How much of the candy bar does each person get?
First, let’s look at this problem visually.
Anne starts with a full candy bar.
Bill breaks off
of the bar.
of what was left.
Dora gets half of the remaining candy bar.
We can see that the candy bar ends up being split four ways, with each person getting an equal amount.
Each person gets exactly
of the candy bar.
We can also examine this problem using rational numbers. We keep a running total of what fraction of the bar remains. Remember, when we read a fraction followed by
in the problem, it means we multiply by that fraction.
We start with 1 bar. Then Bill takes
of it, so there is
of a bar left.
of what’s left, or
of a whole bar. That leaves
of a bar.
That half bar gets split between Anne and Dora, so they each get half of a half bar:
So each person gets exactly
of the candy bar.
If each person’s share is 3 oz, how much did the original candy bar weigh?
Newton’s second law
relates the force applied to a body in Newtons
the mass of the body in kilograms
and the acceleration in meters per second squared
Calculate the resulting acceleration if a Force of
Newtons is applied to a mass of
First, we rearrange our equation to isolate the acceleration,
, dividing both sides by
. Then we substitute in the known values for
The resultant acceleration is
Watch this video for help with the Examples above.
CK-12 Foundation: Solving Real-World Problems Using Multiplication and Division
When multiplying an expression by negative one, remember to multiply the
by negative one.
To multiply fractions, multiply the numerators and multiply the denominators:
The product of two numbers is the same whichever order the items to be multiplied are written.
When three or more numbers are multiplied, the sum is the same regardless of how they are grouped.
Multiplicative Identity Property:
The product of any number and one is the original number.
The multiplication of a number and the sum of two numbers is equal to the first number times the second number plus the first number times the third number.
of a number is the number which produces 1 when multiplied by the original number. The multiplicative inverse of
is the reciprocal
. To find the multiplicative inverse of a fraction, simply
invert the fraction
To divide fractions, invert the divisor and multiply:
An object moving at a certain
will cover a fixed
in a set
. The quantities speed, distance and time are related through the equation
Andrew is driving down the freeway. He passes mile marker 27 at exactly mid-day. At 12:35 he passes mile marker 69. At what speed, in miles per hour, is Andrew traveling?
To find the speed, we need the distance traveled and the time taken. If we want our speed to come out in miles per hour, we’ll need distance in
and time in
The distance is
or 42 miles. The time is 35 minutes, or
hours, which reduces to
. Now we can
the values for distance and time into our equation for speed.
Andrew is driving at 72 miles per hour.
For 1-8, perform the operations of multiplication and division.
For 9-11, solve the real-world problems using multiplication and division.
The label on a can of paint says that it will cover 50 square feet per pint. If I buy a
pint sample, it will cover a square two feet long by three feet high. Is the coverage I get more, less or the same as that stated on the label?
The world’s largest trench digger, "Bagger 288", moves at
mph. How long will it take to dig a trench
Newton force applied to a body of unknown mass produces an acceleration of
. Calculate the mass of the body.