We want to know how many miles of elevation the hiker should gain on the fourth day to gain 14 mile of elevation over the four days. Let's start by looking at the given table!
Day
Change in elevation (miles)
1
- 1/4
2
1/2
3
- 1/5
4
?
We want the change of the elevation over the four days to be equal to 14 mile. In other words, we want the sum of the changes in the elevation on each of the four days to be 14 mile. This means that the following equation should hold true.
- 1/4+ 1/2+( - 1/5)+?=1/4
Notice that the change in the elevation on the fourth day is the difference between 14 mile and the change in the elevation over the three days. Let's write an expression for the change in the elevation on the fourth day!
1/4-( - 1/4+ 1/2+( - 1/5))
Now we can evaluate the expression. When we add or subtract fractions, they should have the same denominator. In this case, we will find the least common denominator of the four fractions with three different denominators. Let's compare the multiples of 2, 4, and 5!
Multiples of2
2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Multiples of4
4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Multiples of5
5, 10, 15, 20, 25, 30, 35, 40, 45, 50
We found that 20 is the least common denominator of the fractions. We will use this information to rewrite each fraction with the same denominator.
Next, we will add two fractions in the parentheses. Recall that the sum of two fractions with different signs should have the sign of the fraction with the greater absolute value. We can start by calculating the absolute value of the fractions.
|- 9/20|=9/20and|10/20|=10/20
The absolute value of 1020 is greater than the absolute value of - 920. This means that our sum will be a positive number. Let's perform the addition!
Now we can rewrite the difference as a sum of 520 and the opposite of 120, - 120.
5/20-1/20 ⇔ 5/20+(- 1/20)
Notice that the addends in our fraction have different signs. Like we did before, we will compare the absolute values of the fractions to determine the sign of their sum.
|5/20|=5/20and|- 1/20|=1/20
The absolute value of 520 is greater than the absolute value of - 120. This means that our sum will be a positive number. We can finally evaluate our expression.
