Word Problems: Rational Numbers
Rational Numbers in the Real World
As a review, rational numbers are numbers that can be written as a ratio of two integers. The picture at right shows examples of rational numbers.
Rational numbers in word problems can be used to represent things in the real world, such as cost, length, volume, etc. Because they represent values in real life, they often come with a unit, such as dollars, meters, etc. We solve word problems the same way as we solve regular equations with rational numbers; find key values, make an equation, and solve.
Example 1. The taxi costs $20.00 plus an additional $2.50 for every mile traveled. You paid $38.75 for the trip from the airport to your hotel. How far away is your hotel from the airport?
Our unknown in this problem is the number of miles traveled. Let's call that x. Since every mile costs $2.50, the cost of the trip without the initial fee is 2.50x. Adding on the initial fee gives us 2.50x + 20.00. We set this equal to the total cost of the trip, which is $38.75. We then solve for x.
Example 2. A basket contains apples, oranges, and bananas. The fruit weighs 58/3 kg. The weight of the apples is 73/9 kg. The weight of the oranges is 38/9 kg. There are 8 bananas. Rounded to the nearest tenth of a kilogram, what is the weight of one banana?
Since the total weight of all the fruit is 58/3 kg, we can subtract the weight of the apples and oranges from the total weight to get the weight of the bananas. We can then divide the weight of the bananas by the number of bananas to get the weight of a single banana.
Word Problems with Volume and Surface Area
As a reminder, volume is the amount of space an object occupies or contains. Surface area is the area of the outside surface of an object.
Volume is in 3D, so the unit of volume would be the unit of length cubed since length is in 1D. The unit for surface area is the unit of length squared because we treat each face of an object as a 2D surface, even when we add together areas on a three-dimensional object. In these word problems, we will mainly be focusing on volume and surface area associated with rectangular prisms and cylinders.
Example 3. The park has a rectangular kids pool with length 18.5 ft, width 76/5 ft, and a depth of 2.5 ft. How much water can the pool hold?
The amount of water swimming pool can hold is representative of the volume of the pool. We calculate the volume of a rectangular prism using V = length x width x height. In this case, the height is the depth of the pool. Before plugging in values, we need to make sure all rational numbers are converted into the same form. In this case, we will convert 76/5 ft into a decimal like the other two dimensions.
Example 4. Fred is building and a picture frame. He will first cut out a square piece of wood with side length 8 inches and a depth of 0.75 inch.
a) What is the volume and surface area of the piece of wood he cuts out?
b) From the center, Fred cuts out a smaller square piece with side length 6.5 in. To the nearest hundredth, how much volume is in the remaining "frame"?
a) What is the volume and surface area of the piece of wood he cuts out?
b) From the center, Fred cuts out a smaller square piece with side length 6.5 in. To the nearest hundredth, how much volume is in the remaining "frame"?
a) We use V = lwh to find the volume of the initial piece of wood and SA = 2(lw+lh+wh) to find surface area. Note that the formula for surface area of a rectangular prism is the sum of the areas of all six sides, and opposite sides in a rectangular prism have the same area.
b) We can find the volume in the remaining picture frame by subtracting the volume of the smaller piece of wood he cuts out from the original volume of the wood.
Example 5. An apartment building needs to clean out their cylindrical water tank. To do so, workers must filter out the water. The base of the tank has a radius of 6.25 m and a height of 7.5 m. Round a) to the nearest hundredth and b) to the nearest whole percent.
a) What is the volume of the filled tank?
b) The water is filtered into a truck with a cylindrical storage compartment. The compartment has a radius of 4.5 m and a length of 15 m. After all the water is filtered, what percent of the volume in the compartment is taken up?
a) What is the volume of the filled tank?
b) The water is filtered into a truck with a cylindrical storage compartment. The compartment has a radius of 4.5 m and a length of 15 m. After all the water is filtered, what percent of the volume in the compartment is taken up?
a) We can find the volume of the cylindrical water tank using the formula V = (pi)(r^2)h, with r = radius and h = height.
b) The fraction of the storage compartment taken up is the volume of the tank divided by the volume of the compartment. We can convert this to a percent by multiplying by 100%.
