How many times the unit quantity

4.OA.A.24.NBT.B.5 · take · grade 4

Archetype: Multiplicative Comparison and Unit Rate · step in a 7-type progression

An airplane travels $864\ \text{km}$ in one hour. If it flies at the same speed for $4$ hours and $30$ minutes, find how many $\text{km}$ it can travel.

Show solution

Understand

An airplane flies 864 km every hour at a steady speed. Find how far it travels in 4 hours and 30 minutes.

Givens

The airplane travels 864 km in one hour
The speed stays the same
The flying time is 4 hours and 30 minutes

Unknowns

The total distance in km

Constraints

30 minutes is half of one hour, so the time is 4.5 hours
Distance equals the per-hour amount times the number of hours

Plan

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

Distance is a rate problem: km-per-hour times hours gives km. Tracking the unit 'km in one hour' as the per-unit amount and recognizing 30 minutes as half an hour lets us split the trip into 4 full hours plus a half hour.

Execute

#7 Identify Subproblems 4.NBT.B.5

Each hour adds 864 km, so 4 hours add 864 x 4 = 3456 km.

864 \times 4 = 3456 \text{ km}

Four equal hours is just multiplying the one-hour distance by 4.

#8 Analyze the Units 4.OA.A.2

30 minutes is half an hour, so the plane covers half of 864 km, which is 864 / 2 = 432 km.

864 \div 2 = 432 \text{ km}

Half an hour means half the hourly distance -- a simple halving.

#7 Identify Subproblems 4.OA.A.3

Total distance is the 4-hour part plus the half-hour part: 3456 + 432 = 3888 km.

3456 + 432 = 3888 \text{ km}

Adding the whole-hour and half-hour distances gives the full trip.

Answer: 3888 km

Review

In 4.5 hours the distance should be a bit more than 4 x 864 = 3456 and less than 5 x 864 = 4320. The answer 3888 falls right between them, as expected for 4.5 hours. Units stay in km throughout.

Convert to half-hours: 4 h 30 min is 9 half-hours, and each half-hour covers 432 km, so 432 x 9 = 3888 km, the same result.

Standards · min grade 4

4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison — Using the per-hour distance as the unit amount and scaling it to find the half-hour distance.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Combining the 4-hour and half-hour distances into the total.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Computing 864 x 4 for the four whole hours.

💡 This only needs Grade 4 multiplying and halving -- 4 full hours plus a half hour of the same speed!