Number Problems

This type of problems may state something about "Finding two numbers...", even while they may be about a certain topic, they give you info that leads to two numbers and the relationship between these numbers. Here are several examples of these this type of application problem:

Example

The tallest snowman ever recorded—really a snow woman named Olympia—was built by residents of Bethel, Maine, and surrounding towns. Her body and head together made up her total record height of 122 ft. The body was 2 ft longer than 14 times the height of the head. What were the separate heights of Olympia's head and body?

Data: Guinness World Records

Solution

1. Define the variables:

$h\:$is the height of Olympia's head and $b\:$is the height of her body.

1. Write the equations:

$b+h=122$

$b=14h+2$

1. Solve the equations:

Since the second equation is already solved for a variable, let's use substitution.

$\left(14h+2\right)+h=122$

Combining like terms

$15h+2=122$

Subtracting 2 from both sides

$15h=120$

Dividing by 15

$h=\frac{120}{15}=8$

Substituting $h=8$ into $b=14h+2$

$b=14h+2=14\left(8\right)+2=112+2=114$

1. Answer in a sentence:

The height of Olympia the snowwoman's head was 8 ft, and her body was 114 ft tall.

Example

In 2016, there were 96 species of birds in the United States that were considered threatened or endangered. The number considered threatened was 1 more than one-fourth of the number considered endangered. How many U.S. bird species were considered endangered or threatened in 2016?

Solution

1. Define the variables:
   • d is the number of endangered bird species and t is the number of threatened bird species in the United States in 2016.
2. Write a system of equations to solve:

t+d=96

t=14d+1

    3. Solve the system of equations:

Since one equation already has a variable isolated, let's use the substitution method:

$t+d=96$

Substituting $\frac{1}{4}d+1$ for $t$

$\frac{1}{4}d+1+d=96$

Combining like terms 

$\frac{5}{4}d+1=96$

 Subtracting 1 from both sides

$\frac{5}{4}d=95$

Multiplying both sides by $\frac{4}{5}$

$d=\frac{4}{5}\cdot95=76$

Next, using either of the original equations, we substitute and solve for t:

$t=\frac{1}{4}\cdot76+1=19+1=20$

   4. Answer in a sentence:

There were 76 bird species considered endangered and 20 considered threatened.

Example

Here is another example from Khan's Academy: