Fraction problem for primary school

Question

I saw a problem yesterday, which can be easily be solved if we are using fractions. But the problem is for the 4th grade children, and I don't know how to solved this using what they what learned.

I tried solved it using the graphic method ( segments ). Here's the problem:

A team of workers has to finish a road. First day, they builded 3/4 of the road and 2 meters, the 2nd day 3/4 of the remaining road and 2 meters. Last day, the remaining length was 1 meter. What was the length of the road?

Answer 1

You can work backwards starting from the last day:

• The work took place over days 1, 2, and 3. They built the complete road starting from nothing.
• They built some amount of the road on day 1.
• They had to build the rest of it on days 2 and 3. We can think about days 2 and 3 collectively.
• On days 2 and 3, they built the rest of it. They built 3/4 of the rest, then they built 3m. This finished the road.
• To finish the road, they had to build 1/4 of the rest. This means that 3m is 1/4 of the rest.
• So 12m is the amount of road they had to build overall on days 2 and 3.
• On day 1, they built 3/4 of the road and 2 meters. When they finished, we know that there were 12m left to build on days 2 and 3.
• So, after they built 3/4 of the road, there were 12m+2m = 14m left.
• After they built 3/4 of the road, there was 1/4 of the road left.
• This means that 14m is 1/4 of the total length of the road.
• So the overall length of the road is 56m.

Answer 2

Time 1: They did $\frac 34$ of the road. $\frac 14$ is left.

Time 1a: they did $2$ meters.

Time 2: They did $\frac 34$ of what was left. What remains is $\frac 14$.

Time 2a: they did $2$ meters.

Time 3: 1 meter is left.

Work backwards:

The $1$ meter of time 3) and the $2$ meters of time 2a) is $3$ meters.

That $3$ meters is the $\frac 14$ that was left at Time 2.

So $3$ meters is $\frac 14$ of what? Of $12$ meters. So that's what they did had left when they started time 2).

At time 1b: they did $2$ meters so that and the $12$ meters is $14$ meters.

That is the $\frac 14$ they had left at time 1.

So $14$ meters is $\frac 14$ of what? Of $4*14= 56$ meters. So $56$ meters is the total work.

Verify:

Day 1: they do $\frac 34$ of $56$ plus $2$ that is $\frac 34*16 + 2 = 42+2 =44$ There is $56 -44 = 12$ meters left.

Day 2: they do $\frac 34$ of $12$ plus $2$ that is $\frac 34*12 + 2 = 11$. There is $12-11= 1 $ meter left.

Day 3: there is $1$ meter left.

Verify a 2nd time just to make sure. Day 1: $44$, Day 2: $11$, Day 3: $1$. $44 + 11 + 1 =56$ meters.