Motion Problems with Fractions
Motion Problems with Fractions applies the distance-rate-time formula (d = rt, or t = d/r) to solve real-world problems involving fractions, as taught in Yoshiwara Elementary Algebra Chapter 8: Algebraic Fractions. Grade 6 students set up equations using a variable for unknown speed, organize given information in a table, and solve rational equations. This topic bridges algebra skills with practical problem-solving contexts like travel and rates.
Key Concepts
Property Motion problems use the formula distance = rate × time ($$d=rt$$). This can be rearranged to $$t = \frac{d}{r}$$. To solve problems, let a variable represent the unknown speed. Use a table to organize distance, rate, and time for each part of the trip. Use the formula $$t = \frac{d}{r}$$ to create expressions for time, then write an equation based on the total time or a comparison of times.
Examples A kayak travels 24 miles upstream and back in 9 hours. The river's current is 2 mph. Find the kayak's speed ($k$) in still water. The equation is $$\frac{24}{k 2} + \frac{24}{k+2} = 9$$. The solution is $$k \approx 5.5$$ mph. A train travels 300 miles with a tailwind of 25 mph and 200 miles back against the same wind. The total travel time is 5 hours. Find the train's speed ($s$). The equation is $$\frac{300}{s+25} + \frac{200}{s 25} = 5$$. The solution is $$s=75$$ mph. Maria jogs 6 miles to the park and then walks 4 miles back home. Her jogging speed is twice her walking speed ($w$). The total trip took 2 hours. Find her walking speed. The equation is $$\frac{6}{2w} + \frac{4}{w} = 2$$. This simplifies to $$\frac{7}{w}=2$$, so $$w=3.5$$ mph.
Explanation For tricky motion problems, the key is often time. By expressing time as distance divided by rate ($$t=d/r$$), you can create fractional equations. Solving these equations reveals the unknown speeds or distances you're looking for.
Common Questions
What formula is used for motion problems?
Motion problems use d = rt (distance equals rate times time). Rearranging gives t = d/r, which is useful when time is the unknown.
How do you set up a motion problem with fractions?
Assign a variable to the unknown speed, then express all distances or times as fractions using d = rt. Create an equation based on the given relationship (such as equal times or total distance) and solve.
Where are motion problems with fractions covered in Yoshiwara Elementary Algebra?
This topic is in Chapter 8: Algebraic Fractions of Yoshiwara Elementary Algebra, where algebraic fraction techniques are applied to real-world scenarios.
Why do motion problems involve fractions?
When you divide distance by rate (t = d/r), the result is often a fraction. Problems involving two travelers or two legs of a journey naturally produce fractional expressions.
What is the most common mistake in motion problems?
Mixing up which quantity is unknown. Always clearly define your variable, set up a table with distance, rate, and time columns, and check that your equation correctly represents the problem conditions.