Bigger units express numbers more simply

4.MD.A.14.MD.A.2 · adapt · grade 4

Archetype: Length as Sum of Parts with Unit Matching · step in a 7-type progression

A fence is made by joining $78$ rods, each $27\ \text{in}$ long, end to end without overlapping. Find the length of the fence in feet and inches. (Use $12\ \text{in} = 1\ \text{ft}$ .)

Show solution

Understand

A fence is built by joining 78 rods end to end, each 27 inches long, with no overlap. Find the total length expressed in feet and inches, using 12 in = 1 ft.

Givens

There are 78 rods
Each rod is 27 in long
Rods are joined end to end with no overlap
12 in = 1 ft

Unknowns

The total fence length in feet and inches

Constraints

Total length in inches is 78 times 27
To convert inches to feet, divide by 12; the remainder is leftover inches (less than 12)

Plan

#8 Analyze the Units · also uses: #7 Identify Subproblems

This is a measurement-conversion problem: first find the total inches by multiplying, then convert to the larger unit of feet by dividing by 12, keeping the remainder as inches. Watching the units (inches to feet) guides the divide-by-12 step.

Execute

#7 Identify Subproblems 4.NBT.B.5

Joining 78 rods of 27 in each gives a total of 78 x 27 = 2106 in.

78 \times 27 = 2106 \text{ in}

End-to-end with no overlap just means add up all the rod lengths, which is one multiplication.

#8 Analyze the Units 4.MD.A.1

Divide the total inches by 12 because 12 in make 1 ft. 2106 / 12 = 175 with a remainder of 6, so the length is 175 ft and 6 in.

2106 \div 12 = 175 \text{ R } 6 \Rightarrow 175 \text{ ft } 6 \text{ in}

Grouping every 12 inches into a foot is exactly division with a remainder of leftover inches.

Answer: 175 ft 6 in

Review

Convert back: 175 ft = 175 x 12 = 2100 in, plus 6 in = 2106 in, matching the total. The leftover 6 in is less than 12, so the conversion is proper.

Identify subproblems (tool 7): convert each rod first, since 27 in = 2 ft 3 in, then 78 rods give 156 ft and 78 x 3 = 234 in = 19 ft 6 in, and 156 + 19 = 175 ft 6 in, the same answer.

Standards · min grade 4

4.MD.A.1 Know relative sizes of measurement units and convert larger to smaller units — Converting the total inches into feet and inches using 12 in = 1 ft.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying 78 rods by 27 in to get the total inches.

💡 This only needs Grade 4 multiplying and dividing by 12 -- add the rods, then bundle every 12 inches into a foot!