PSHS NCE Rates, Work, and Distance Problems Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Rates, Work, and Distance Problems

Use the right relationship quickly: distance = rate × time, unit rate, and simple work-rate thinking.

Grade 6–7 friendlyRate × TimeRegular practiceQuick Fire + Mastery

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Rates, Work, and Distance Problems

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Rate tells how much happens in one unit of time

A cyclist travels 60 km in 3 hours.

60 ÷ 3 = 20 km/h

The rate is 20 km per hour.

Three useful relationships:
Distance = Rate × Time
Rate = Distance ÷ Time
Time = Distance ÷ Rate

DO IT FAST

Use WHAT DO I NEED?

1. Need distance? Multiply rate × time.
2. Need rate? Divide distance ÷ time.
3. Need time? Divide distance ÷ rate.
4. Need work per hour? Think “fraction of the job completed in one hour.”

Fast clue: If the problem gives “per hour,” “per minute,” or “each,” you are probably dealing with a rate.

WORKED EXAMPLES

Six patterns you should recognize

1. Find distance

A car travels at 50 km/h for 4 hours.

50 × 4 = 200 km

200 km

2. Find rate

A runner covers 18 km in 3 hours.

18 ÷ 3 = 6 km/h

6 km/h

3. Find time

A bus travels 180 km at 60 km/h.

180 ÷ 60 = 3 hours

3 hours

4. Unit rate

5 notebooks cost ₱150.

₱150 ÷ 5 = ₱30 each

₱30 per notebook

5. Simple work rate

A machine can finish a job in 4 hours.

In one hour it finishes 1/4 of the job.

Work rate = 1/4 job per hour

6. Combined work

Machine A does 1/4 of a job per hour. Machine B does 1/4 per hour.

1/4 + 1/4 = 1/2 job per hour

Together they finish in 2 hours.

TRY WITH ME

Pick the missing quantity

A van travels 240 km at 60 km/h. How long does the trip take?

1. We know distance and rate.
2. We need time.
3. Time = distance ÷ rate.
4. 240 ÷ 60 = 4.

4 hours

COMMON TRAPS

Watch out for these

Multiplying when you should divide.
Ask first: am I looking for distance, rate, or time?

Ignoring units.
Hours and minutes must match.

Treating work time like ordinary addition.
If one worker takes 4 hours, the rate is 1/4 job per hour.

Missing “each” or “per.”
Those words usually signal unit rate.

REGULAR PRACTICE

Build the skill

These mix distance-rate-time, unit rates, and simple work-rate questions.

QUICK FIRE

Can you choose the right operation fast?

Five short questions. Decide whether to multiply or divide before calculating.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Rate problems still slowing you down?

Send Teacher Abi the problem that confused you and your result. I can help you identify what the problem is really asking for.

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PSHS NCE REVIEW PATH

Choose your next step

Next competency: Geometry and Measurement

Previous: Algebraic Expressions and Simple EquationsPSHS NCE Student Hub

PSHS NCE Ratios and Proportions Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Ratios and Proportions

Compare quantities, build equivalent ratios, and solve proportion problems without guessing.

Grade 6–7 friendlyEquivalent ratiosRegular practiceQuick Fire + Mastery

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A ratio compares two quantities

A basket has 6 red balls and 9 blue balls.

red : blue = 6 : 9 = 2 : 3

The ratio simplifies just like a fraction.

Order matters. Red to blue is 2:3. Blue to red is 3:2.

DO IT FAST

Use MATCH → SCALE → CHECK

1. MATCH: Keep the quantities in the same order.
2. SCALE: Multiply or divide both parts by the same number.
3. CHECK: Make sure both sides of the proportion changed by the same factor.

Example: 3:5 = 6:10 because both parts were multiplied by 2.

WORKED EXAMPLES

Six patterns you should recognize

1. Simplify a ratio

12 : 18

Divide both by 6.

2 : 3

2. Build an equivalent ratio

4:7 = ? : 21

7 became 21, so multiply by 3.

4 × 3 = 12

12 : 21

3. Find a missing term

5:8 = 15:x

5 became 15, so multiply by 3. Then 8 × 3 = 24.

x = 24

4. Part-to-whole

A class has 8 boys and 12 girls. What is the ratio of boys to all students?

8 : 20 = 2 : 5

2 : 5

5. Rate as a ratio

A car travels 180 km in 3 hours.

180 ÷ 3 = 60 km/h

60 km per hour

6. Use a proportion in context

3 notebooks cost ₱75. At the same rate, how much do 5 notebooks cost?

One notebook costs ₱25, so 5 cost ₱125.

₱125

TRY WITH ME

Find the missing number

4 : 6 = 10 : x

1. 4 became 10. That is ×2.5.
2. Do the same to 6.
3. 6 × 2.5 = 15.

x = 15

COMMON TRAPS

Watch out for these

Switching the order.
If the question asks cats to dogs, do not answer dogs to cats.

Changing only one part.
Equivalent ratios must scale both parts by the same factor.

Confusing part-to-part with part-to-whole.
Read what the ratio is actually comparing.

Cross-multiplying too soon.
If the scale factor is obvious, use it. It is often faster.

REGULAR PRACTICE

Build the skill

These mix simplifying ratios, equivalent ratios, missing terms, rates, and short word problems.

QUICK FIRE

Can you scale it fast?

Five short questions. Use the easiest scale factor you can see.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Ratios still getting mixed up?

Send Teacher Abi the problem that confused you and your result. I can help you spot the easiest scale factor.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Percent

Previous: Estimation and ReasonablenessPSHS NCE Student Hub

Distance Rate Time Problems UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Distance–Rate–Time Problems

Build one distance equation for each traveler, then connect the distances using the story.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Distance–Rate–Time Problems

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Build one distance equation, then connect the travelers

The core relationship is d=rt, but UPCAT-style questions usually hide which distance, rate, or time belongs in the equation.

distance = rate × timerate = distance ÷ timetime = distance ÷ rate

Convert units before calculating. For an entire trip, average speed is total distance ÷ total elapsed time, including stops. When travelers move toward each other, add their speeds. For catch-up in the same direction, subtract their speeds.

Draw the distance or gap

Mark starting points, direction, speed, and any delayed departure.

Use relative speed carefully

Towardopposite directions usually add; catching up in one direction uses a difference.

DO IT FAST

Decide what happens to the gap

Problem: Two vehicles are 180 km apart and move toward each other at 50 kmh and 40 kmh.

Recognize: Every hour, the first closes 50 km of the gap and the second closes 40 km. Together they close 90 km per hour.

time = 180(50+40) = 2 hours

For a same-direction catch-up, use the faster speed minus the slower speed because only that difference reduces the lead.

Why it works

Relative speed measures how quickly the distance between travelers changes. Adding or subtracting rates is justified by whether both motions reduce the gap or only one traveler gains on the other.

WORKED EXAMPLES

Five forms you should recognize

1. Basic distance

A van travels at 60 kmh for 2.5 hours.

d=rt=60(2.5)=150 km

2. Average speed with a stop

A trip covers 150 km in 2.5 hours of driving and includes a 30-minute stop. Total elapsed time is 3 hours:

average speed=1503=50 kmh

3. Meeting from opposite towns

Two vehicles 240 km apart approach at 50 kmh and 70 kmh.

closing speed=50+70=120 kmhtime=240120=2 hours

4. Catch-up after a head start

A slower vehicle has a 45-km lead at 60 kmh; a faster one follows at 75 kmh.

gain speed=75−60=15 kmhtime=4515=3 hours

5. Equal-distance round trip

A car travels 120 km at 60 kmh and returns 120 km at 40 kmh.

total distance=240 kmtotal time=12060+12040=2+3=5 haverage speed=2405=48 kmh

Do not simply average 60 and 40 because the car spends different amounts of time at the two speeds.

COMMON TRAPS

Check before you commit

• Mixing minutes with hours
• Averaging speeds instead of using total distance over total time
• Ignoring stopped time
• Adding speeds in a same-direction catch-up
• Subtracting speeds when travelers approach each other
• Finding meeting time but not the requested meeting location

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Distance–Rate–Time Problems FAQ

Why are there two mastery sets?

An unsuccessful attempt loads a different five-question set so memorizing one set does not verify mastery.

Does 5/5 mean permanent mastery?

It verifies the competency today. Revisit it later through mixed practice and simulations.

RELATED COMPETENCIES

Continue your mathematics review.

Ratio and ProportionLinear EquationsWork-Rate Problems

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