Examkin

Question

Difficulty: MediumSolving Linear Equations

A marathon runner plans to run a total of dd miles this week. The runner schedules 23(d−6)\frac{2}{3}(d - 6) miles of the run on soft dirt trails and 0.4(d+15)0.4(d + 15) miles on asphalt roads. If the runner completes only these two segments for a total of 3434 miles, what is the value of dd?

Answer: 30 miles

Answer

The value of dd is 30.
The value of dd that satisfies the equation is 3030 because substituting 3030 back into the original equation yields a true statement: 23(30−6)+0.4(30+15)=16+18=34\frac{2}{3}(30 - 6) + 0.4(30 + 15) = 16 + 18 = 34.

Step-by-Step Solution

1
Set up the equation based on the given information: the trail running distance plus the road running distance equals the total distance.
23(d−6)+0.4(d+15)=34\frac{2}{3}(d - 6) + 0.4(d + 15) = 34
This relates the individual segment distances to the total planned mileage of 3434 miles.
2
Distribute the coefficients to eliminate the parentheses.
23d−4+0.4d+6=34\frac{2}{3}d - 4 + 0.4d + 6 = 34
Distributing 23\frac{2}{3} to (d−6)(d-6) gives 23d−4\frac{2}{3}d - 4, and distributing 0.40.4 to (d+15)(d+15) gives 0.4d+60.4d + 6.
3
Combine the constant terms on the left side of the equation.
23d+0.4d+2=34\frac{2}{3}d + 0.4d + 2 = 34
Combining the constants −4-4 and 66 yields 22.
4
Subtract 2 from both sides of the equation to isolate the variable terms.
23d+0.4d=32\frac{2}{3}d + 0.4d = 32
Subtracting 22 from both sides simplifies the equation to have variable terms on one side and constant terms on the other.
5
Convert the decimal 0.40.4 to a fraction to combine the coefficients of dd.
23d+25d=32\frac{2}{3}d + \frac{2}{5}d = 32
Converting 0.40.4 to 410=25\frac{4}{10} = \frac{2}{5} allows us to work with a common denominator.
6
Find a common denominator to add the fractions.
1015d+615d=32⇒1615d=32\frac{10}{15}d + \frac{6}{15}d = 32 \Rightarrow \frac{16}{15}d = 32
The least common multiple of 33 and 55 is 1515.
7
Multiply both sides of the equation by the reciprocal of the coefficient of dd to solve for dd.
d=32×1516⇒d=2×15⇒d=30d = 32 \times \frac{15}{16} \Rightarrow d = 2 \times 15 \Rightarrow d = 30
Multiplying by 1516\frac{15}{16} isolates dd on the left side of the equation.

Key Concept

Solving linear equations involving fractions and decimals by distributing, combining like terms, and isolating the variable.

Alternative Method

Instead of converting the decimal 0.40.4 to a fraction, you can multiply the entire equation by a common multiple like 1515 to eliminate both the fraction and the decimal: 15[23(d−6)+0.4(d+15)]=15(34)⇒10(d−6)+6(d+15)=510⇒10d−60+6d+90=510⇒16d+30=510⇒16d=480⇒d=3015 \left[ \frac{2}{3}(d - 6) + 0.4(d + 15) \right] = 15(34) \Rightarrow 10(d - 6) + 6(d + 15) = 510 \Rightarrow 10d - 60 + 6d + 90 = 510 \Rightarrow 16d + 30 = 510 \Rightarrow 16d = 480 \Rightarrow d = 30. This method avoids working with fractional coefficients.
Estimated Time:1m 30s
Rate this question

Topics

Solving Linear Equations

More questions from this topic