Length conversions using dimensional analysis

Multi-step chain. Pick the right factors so the unwanted units cancel and only the target unit survives.

01 Mini-Lesson Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here's an example of what you'll see.

Bikeathon distance v1

As part of an annual fundraiser, Ann joined a bikeathon. The track she biked on was 1,870 yd long. Ann biked 32 laps. How many miles did she bike?

ALEKS gives you a bank of ratios. Pick three and place them in the equation so units cancel down to miles.

▸ Ratios available

3 ft
1 yd

1 yd
3 ft

1870 yd
1 lap

1 lap
1870 yd

5280 ft
1 mi

1 mi
5280 ft

32 laps×××=miles

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 How to Solve It

laps → yd → ft → mi

Three factors. Each one cancels a unit and leaves the next link's setup. Pick the version of each ratio that puts the unwanted unit on the bottom.

Plan the chain.

Start: laps. Target: miles. There's no direct laps → miles factor in the bank. So plan a path through other units. Track length is in yards and miles relate to feet, so the chain is laps → yd → ft → mi.

Pick three ratios. Cancel along the chain.

From the bank: 1870 yd / 1 lap (cancels laps), 3 ft / 1 yd (cancels yards), 1 mi / 5280 ft (cancels feet, leaves miles).

32 laps \times 1870 yd 1 lap \times 3 ft 1 yd \times 1 mi 5280 ft

All the middle units cancel. Only miles survives.

Multiply across.

32 \times 1,870 \times 3 / 5,280 = 34 mi

179,520 ÷ 5,280 = 34. Ann biked 34 miles.

▸ COMMON SLIPS(1) Wrong-direction factor. Picking 1 yd / 3 ft instead of 3 ft / 1 yd puts yards on top, which doesn't cancel anything. Always check: the unit you want to cancel goes on the BOTTOM of the next factor. (2) Skipping a step. Going laps → mi directly without the intermediate yd/ft chain leaves you stuck. Plan the full path before you start placing ratios.

04 Practice — Step by Step

David walked 20 laps on a 264 yd track. Walk the same chain to find his total distance in miles. Use 1 yd = 3 ft and 1 mi = 5280 ft.

Step 1: laps → yards.

David walked 20 laps on a 264 yd track. How many yards did he walk in total?

yards =

Step 2: yards → feet.

Convert 5,280 yards to feet. Use 1 yd = 3 ft.

feet =

Step 3: feet → miles.

Convert 15,840 feet to miles. Use 1 mi = 5,280 ft.

miles =

▸ NICE WORK

You walked a 3-factor chain end to end.

Same recipe Ann used, just different numbers. Plan the chain, pick the version of each ratio that cancels the previous link's unit, then multiply across. ALEKS varies the laps, the track length, and even the units (yd vs. ft starting points). The move stays the same. If you can do this one, the Cancellation Sandbox in Hard mode is your friend.

Q2 Q4