May is National Bike Month, so here are a few problems related to bicycles.
Brenden is riding an old-time bicycle that has a back wheel with a diameter of 1 foot and a front wheel with a radius of 1.5 feet. After traveling 1000 feet, how many more rotations than the front tire has the back tire completed? Express your answer to the nearest whole number.
1000 ft/[(0.5)(2)(π)] = 318.309886 rotations of the back wheel
1000 ft/[(1.5)(2)(π)] = 106.103295 rotations of the front wheel
318.309886 – 106.103295 = 212 rotations to the nearest full rotation
Jermaine starts riding his bike at a rate of 10 miles per hour. After 5 minutes, Nelson starts riding from the same starting point as Jermaine and along the same route at a rate of 12 miles per hour. If each boy rides at a constant rate, how many minutes will Jermaine ride his bike for before Nelson catches up with him?
Let’s call the time that both boys are riding t. We know that once Nelson catches up to Jermaine, the two boys will have each ridden the same distance. Let’s call that distance d. Thus, for Jermaine, we can say 10 = d/(t + 5/60) and for Nelson, we can say 12 = d/t. Let’s set both equations equal to d.
d = 10(t + 5/60) and d = 12t → 10(t + 5/60) = 12t
Now, we can solve for t.
10t + 5/6 = 12t → 5/6 = 2t → t = 5/12 hours
We’re asked for an answer in minutes, so (5/12)(60) = 25 minutes. Therefore, Jermaine rode for 25 + 5 = 30 minutes.
At Your Town Bike Shop, handle bar streamers are sold in packs of two and beads for spokes are sold in sets of 10. Georgina purchases 2 packs of handle bar streamers and 3 sets of beads for $27.85 before tax. Alexis purchases 1 pack of streamers and 4 sets of beads for $26.30 before tax. If Nana wants to buy 3 packs of streamers and 6 sets of beads, how much will her bill be before tax?
Let’s let h represent the price of one handle bar streamers pack and b represent the price of one set of beads. Since Georgina purchases 2 packs of handle bar streamers and 3 sets of beads for $27.85, we can say that 2h + 3b = 27.85. Since Alexis purchases 1 pack of streamers and 4 sets of beads for $26.30, we can also say that h + 4b = 26.30.
With two equations and two variables, we can solve the system of equations:
h + 4b = 26.30
2h + 3b = 27.85
By solving the top equation for h, we can use substitution to solve for b.
h = 26.30 – 4b
2(26.30 – 4b) + 3b = 27.85
52.60 – 8b + 3b = 27.85
5b = 24.75
b = 4.95
Substituting this value for b into one of the equations will allow us to solve for h:
h + 4(4.95) = 26.30
h = 26.30 – 19.80
h = 6.50
Thus, 3 packs of streamers and 6 sets of beads cost 3(6.50) + 6(4.95) = 19.50 + 29.70 = $49.20.
♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦