Draw both cars on the same road.
Same direction means one lane chases the other.
Why this one subtracts
Both cars move the same way, so the gap is only the extra distance Car A has covered — a difference, not a total.
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Distance Apart
Two vehicles, same direction — the faster one pulls ahead.
The problem
Two vehicles leave the same point at the same time, going the same way. Car A travels at 75 mph and Car B at 55 mph. How long until they are 100 miles apart?
Same direction means the gap opens at the difference: 75 − 55 = 20 mph, so 100 ÷ 20 = 5 hours. Adding the speeds to get 130 is the opposite-direction move — the direction word decides whether you add or subtract.
Fill a speed × time = distance table.
One row per car. Put the faster car on top.
| Vehicle | Speed | Time | Distance |
|---|---|---|---|
| Car A | 75 | t | 75t |
| Car B | 55 | t | 55t |
distance = speed · time, so 75 · t and 55 · t
One clock, not two
They left together, so the same t sits in both rows. If one had a head start you would need t and t − 1 instead.
Subtract the shorter trip from the longer one.
D₁ − D₂ = Dapart
75t − 55t = 100
Bigger distance first. Writing 55t − 75t would give −100 and a negative time.
Combine the two t terms.
75t − 55t = 100
75t and 55t are like terms — subtract the numbers only.
What 20 actually means
75 mph − 55 mph = 20 mph
Car A gains 20 miles on Car B every hour.
This 20 mph is the gap speed. Opposite directions would have added the speeds instead.
Solve for t.
20t = 100
from Step 4
t = 5
÷ 20 both sides
Check your units
miles ÷ mph = hours
A 20 mph gap needs 5 hours to reach 100 miles. Slower gap, longer wait — the answer should feel large.
Read what is asked once t is known. How far has Car B gone? Put t into its row: 55 × 5 = 275 miles from the start. At what time? Add t to the start: cars leaving at 7:00 a.m. are 100 miles apart at 12:00 noon.
A staggered start: give each one its own time.
The early vehicle has been driving longer.
truck: 4 h × 40 = 160 km
1 + 3 hours
car: 3 h × 64 = 192 km
3 hours
192 − 160 = 32 km
the car is now ahead
Do not drop the head start
Using the speed gap alone, 24 × 3 = 72 km, ignores the 40 km the truck had already gone. 72 − 40 = 32 agrees with the table.
Variations: minutes, a speed ratio, a gap at the start.
The difference of the speeds still drives each one.
Hours and minutes: two vans leave together going the same way at 52 and 64 km/h. How far apart are they after 1 h 15 min?
1 h 15 min = 1¼ h = 54 h
15 min is a quarter hour
(64 − 52) × 54 = 12 × 54 = 15 km
gap speed × time
1 h 15 min is 1.25 h, not 1.15 h: minutes are sixtieths, not hundredths.
A speed ratio: one car is twice as fast as the other. Leaving together the same way, they are 105 km apart after 3 hours. Find both speeds.
2r × 3 − r × 3 = 105
r is the slower speed
3r = 105, so r = 35
6r − 3r
35 km/h and 70 km/h
check: 210 − 105 = 105 ✓
The gap speed is 2r − r = r, so here the gap speed equals the slower car’s speed.
A gap at the start: a car at 70 km/h starts 20 km behind a car at 50 km/h, both heading the same way. When is the faster car 30 km ahead?
20 + 30 = 50 km to gain
close the gap, then lead by 30
70t − 50t = 50, so 20t = 50
gains 20 km each hour
t = 2.5 h
check: 175 km vs 20 + 125 = 145 km, 30 ahead ✓
20t = 30 forgets the 20 km
That gives 1.5 h, when the fast car is only 10 km ahead. A car that starts behind must first make up the starting gap.
Sum and difference of the speeds.
The gap gives the difference; the question gives the sum.
difference: 30 ÷ 2 = 15 km/h
the gap grows at the difference
faster: (135 + 15) ÷ 2 = 75
half of sum + difference
slower: (135 − 15) ÷ 2 = 60
half of sum − difference
Check: 75 + 60 = 135 and (75 − 60) × 2 = 30 ✓.
On the exam
Same-direction items turn on the difference of the speeds — the gap grows, or a chaser gains, at that rate. Check for a head start and for minutes to convert; if both speeds are asked, take half of (sum ± difference).
Check: put 5 hours back into both trips.
Car A
75 · 5 = 375 miles
in 5 hours
Car B
55 · 5 = 275 miles
in 5 hours
Now you try
Practice problem
The problem
Two trucks leave the same depot at the same time going the same way. The faster one travels at 68 mph. After 3 hours they are 54 miles apart. How fast is the slower truck?
The gap opens at 54 ÷ 3 = 18 mph, and that 18 is the difference in their speeds. So the slower truck does 68 − 18 = 50 mph. C is the difference itself, mistaken for a speed.
Hint: the missing number is the slower speed, so it sits in the term you subtract. Keep the minus sign attached to it.