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Arithmetic Reasoning

Distance, Work, Averages & Multi-Step Problems

These Arithmetic Reasoning questions connect several pieces of information involving distance, rate, time, work, productivity, averages, ages, capacity, packages, and multi-step totals. Identify the goal, write down what you know, find one missing piece at a time, keep units consistent, and check the result.

Section 1

Distance, Rate & Time

Distance problems connect distance, rate, and time. Use distance = rate × time, rate = distance ÷ time, and time = distance ÷ rate. Always check the units.

If rate and time are known, multiply to find distance. If distance and time are known, divide to find rate. If distance and rate are known, divide to find time.

Example 1

A truck travels at 48 miles per hour for 5 hours. How far does it travel?

Solution

  1. 1Use distance = rate × time.
  2. 2Substitute: d = 48 × 5.
  3. 3Multiply: d = 240.
Answer: 240 miles

Why: The truck travels 48 miles during each of five hours, so rate multiplied by time gives total distance.

Example 2

A car travels 210 miles in 3.5 hours. What is its average rate?

Solution

  1. 1Use rate = distance ÷ time.
  2. 2Substitute: rate = 210 ÷ 3.5.
  3. 3Divide: rate = 60.
Answer: 60 miles per hour

Why: The total distance is spread across 3.5 hours, so distance divided by time gives miles per hour.

Example 3

A bus must travel 270 miles at an average rate of 54 miles per hour. How long will the trip take?

Solution

  1. 1Use time = distance ÷ rate.
  2. 2Substitute: time = 270 ÷ 54.
  3. 3Divide: time = 5.
Answer: 5 hours

Why: Dividing total distance by miles traveled each hour tells how many hours are required.

Section 2

Elapsed Time & Trips with More Than One Part

Travel problems may involve minutes and hours, stops, different speeds, or several sections. Convert units before combining values and solve each section separately. Do not simply average different speeds unless the problem allows it.

Remember 60 minutes = 1 hour, 30 minutes = 0.5 hour, 15 minutes = 0.25 hour, and 45 minutes = 0.75 hour. For a trip with several parts, find each distance or time, then combine the results.

Example 1

A train leaves at 8:35 AM and arrives at 11:20 AM. How long is the trip?

Solution

  1. 18:35 to 9:35 is 1 hour.
  2. 29:35 to 10:35 is 1 hour.
  3. 310:35 to 11:20 is 45 minutes.
  4. 4Add the parts: 2 hours 45 minutes.
Answer: 2 hours 45 minutes

Why: Elapsed time measures the time between the starting and ending times.

Example 2

A driver travels at 50 miles per hour for 2 hours, then travels another 45 miles. How far is the total trip?

Solution

  1. 1Find the first distance: distance = rate × time = 50 × 2 = 100 miles.
  2. 2Add the second part: 100 + 45 = 145.
Answer: 145 miles

Why: The first distance must be calculated before the two sections can be combined.

Example 3

A driver travels 120 miles at 60 miles per hour, then 90 miles at 45 miles per hour. How much driving time is required?

Solution

  1. 1First section: 120 ÷ 60 = 2 hours.
  2. 2Second section: 90 ÷ 45 = 2 hours.
  3. 3Add driving times: 2 + 2 = 4.
Answer: 4 hours

Why: Each section has a different rate, so calculate each travel time separately before adding.

Section 3

Work, Productivity & Production Rates

Work problems describe how quickly workers complete tasks or machines produce items. The reasoning is the same as other rate problems: production rate = output ÷ time and total production = rate × time.

Independent workers or machines can often have their rates added when they operate at constant rates at the same time. This applies only when the problem says or implies that they work independently.

Example 1

A machine produces 360 bolts in 6 hours. How many bolts does it produce per hour?

Solution

  1. 1Find the unit rate: 360 ÷ 6 = 60.
Answer: 60 bolts per hour

Why: Dividing total production by hours gives production in one hour.

Example 2

A worker packages 28 boxes per hour. How many boxes can the worker package in 7 hours?

Solution

  1. 1Use rate × time.
  2. 228 × 7 = 196.
Answer: 196 boxes

Why: The worker completes 28 boxes during each hour, so repeat that rate seven times.

Example 3

Machine A produces 18 parts per hour and Machine B produces 22 parts per hour. If both run for 5 hours, how many parts do they produce?

Solution

  1. 1Combine hourly rates: 18 + 22 = 40 parts per hour.
  2. 2Multiply by time: 40 × 5 = 200.
Answer: 200 parts

Why: The machines work at the same time, so independent hourly rates can be combined first.

Section 4

Averages

The arithmetic mean, or average, is sum of values ÷ number of values. If an average is known, total = average × number of values.

For a missing value, find the required total first, add the values you know, and subtract. In weighted situations, work with totals rather than blindly averaging two numbers.

Example 1

A student receives scores of 78, 84, 90, and 88. What is the average?

Solution

  1. 1Add: 78 + 84 + 90 + 88 = 340.
  2. 2There are 4 scores.
  3. 3Divide: 340 ÷ 4 = 85.
Answer: 85

Why: The mean divides the total of all values by the number of values.

Example 2

Five test scores have an average of 82. Four are 75, 80, 85, and 90. What is the fifth score?

Solution

  1. 1Required total: 82 × 5 = 410.
  2. 2Known total: 75 + 80 + 85 + 90 = 330.
  3. 3Subtract: 410 - 330 = 80.
Answer: 80

Why: The average gives the total of all five scores; subtracting known scores reveals the missing value.

Example 3

A worker produces 20 units Monday, 28 Tuesday, 24 Wednesday, and 32 Thursday. What is average daily production?

Solution

  1. 1Add: 20 + 28 + 24 + 32 = 104.
  2. 2There are 4 days.
  3. 3Divide: 104 ÷ 4 = 26.
Answer: 26 units per day

Why: Total production is distributed equally across the four days.

Section 5

Age, Total & Comparison Problems

Translate relationships such as older than, younger than, more than, less than, twice as many, three times as much, and combined total before calculating.

"More than" usually means add, "less than" usually means subtract, and "times as many" means multiply. If a total and one part are known, subtract to find the missing part.

Example 1

Jordan is 6 years older than Taylor. Taylor is 17. How old is Jordan?

Solution

  1. 1"Older" means 6 more.
  2. 217 + 6 = 23.
Answer: 23 years old

Why: Six years older means Jordan's age is six more than Taylor's age.

Example 2

A warehouse has three times as many small boxes as large boxes. There are 14 large boxes. How many small boxes?

Solution

  1. 1Multiply the original quantity by 3.
  2. 23 × 14 = 42.
Answer: 42 small boxes

Why: "Three times as many" means multiply the known quantity by three.

Example 3

Two storage rooms contain 96 boxes altogether. The first has 58. How many are in the second?

Solution

  1. 1The combined total is 96.
  2. 2Subtract the known part: 96 - 58 = 38.
Answer: 38 boxes

Why: If the total and one part are known, subtract the known part to find the missing part.

Section 6

Capacity, Packages & Harder Multi-Step Problems

Harder AR problems often connect several basic steps. The calculations may be simple; the challenge is knowing what the final number means, especially when whole containers are required.

Capacity = number of groups × amount per group. If every item must fit in whole containers, leftover items require another container. This differs from asking how many complete groups can be formed.

Example 1

A theater has 18 rows with 24 seats each. If 389 seats are occupied, how many are empty?

Solution

  1. 1Find total capacity: 18 × 24 = 432.
  2. 2Subtract occupied seats: 432 - 389 = 43.
Answer: 43 empty seats

Why: Total capacity must be found before determining how much remains unused.

Example 2

A company must ship 127 items. Each box holds 20. What is the minimum number of boxes needed?

Solution

  1. 1Divide: 127 ÷ 20.
  2. 2Six boxes hold 120, leaving 7 items.
  3. 3The leftovers require one more box: 6 + 1 = 7.
Answer: 7 boxes

Why: The question asks for boxes for all items, so leftover items require another whole box.

Example 3

A warehouse receives 15 crates with 18 items each. It ships 74 items. The rest are placed equally on 7 shelves. How many per shelf?

Solution

  1. 1Original total: 15 × 18 = 270.
  2. 2Subtract shipped items: 270 - 74 = 196.
  3. 3Divide among shelves: 196 ÷ 7 = 28.
Answer: 28 items per shelf

Why: This requires three steps: find the original total, subtract what was removed, and divide the remainder equally.

Quick Review

Distance, Rate & Time

  • Distance: d = rt. Rate: r = d/t. Time: t = d/r. Keep units consistent.

Elapsed Time & Trips

  • Convert minutes and hours when necessary. Solve each trip section separately, then combine results.

Work & Productivity

  • Rate = output/time. Total output = rate × time. Independent rates may be combined.

Averages

  • Mean = sum/number of values. Known average gives total = average × number of values.

Ages & Comparisons

  • More/older means add; less/younger means subtract; times as many means multiply.

Capacity & Packages

  • Find capacity first when necessary. If all items must fit into whole containers, leftovers require another container.

Arithmetic Reasoning Strategy

  • Find the goal, write known information, match units, solve one piece at a time, interpret the result, and check.