Math  /  Algebra

Question39. А хотоос В хот орохоор автобус гарсны дараа 1 цаг болоод хойноос нь машин гарч, автобусы гүйцээд түүнэзс 20 минутын түруүнд В хотод иржээ. Хэрэв хоёр хотын хоорондох зай 280 км бөгөөд машины хурдыг автобусны хурдтай харьцуулахад 3:23: 2 бол тус бүрийн хурдыг ол.

Studdy Solution

STEP 1

What is this asking? A bus and a car travel between two cities.
The car leaves an hour later, overtakes the bus, and arrives 20 minutes earlier.
Knowing the distance and the ratio of their speeds, we need to find the speed of each vehicle. Watch out! Make sure to convert all time units to hours to match the distance units (km).
Don't mix up the times for the bus and the car!

STEP 2

1. Set up variables and relationships
2. Calculate the car's travel time
3. Calculate the bus's travel time
4. Express speeds in terms of variables
5. Set up an equation based on the distance
6. Solve for x
7. Calculate the speeds

STEP 3

Let xx be the travel time of the car in hours.

STEP 4

Since the car left 1 hour later and arrived 20 minutes (or 13\frac{1}{3} of an hour) earlier, the bus's travel time is x+1+13=x+43x + 1 + \frac{1}{3} = x + \frac{4}{3} hours.

STEP 5

The ratio of the car's speed to the bus's speed is given as 3:23:2.

STEP 6

Since the distance is the same, the ratio of their travel times is the inverse of the ratio of their speeds, which is 2:32:3.

STEP 7

So, xx+43=23\frac{x}{x + \frac{4}{3}} = \frac{2}{3}.

STEP 8

3x=2(x+43)3x = 2(x + \frac{4}{3})

STEP 9

3x=2x+833x = 2x + \frac{8}{3} x=83x = \frac{8}{3} hours.
This is the **car's travel time**.

STEP 10

The bus's travel time is x+43=83+43=123=4x + \frac{4}{3} = \frac{8}{3} + \frac{4}{3} = \frac{12}{3} = 4 hours.

STEP 11

Let the car's speed be 3v3v km/h.

STEP 12

Then, the bus's speed is 2v2v km/h.

STEP 13

Distance = Speed \cdot Time 280=(3v)(83)280 = (3v)(\frac{8}{3})

STEP 14

280=8v280 = 8v v=2808=35v = \frac{280}{8} = 35

STEP 15

The car's speed is 3v=335=1053v = 3 \cdot 35 = \textbf{105} km/h.

STEP 16

The bus's speed is 2v=235=702v = 2 \cdot 35 = \textbf{70} km/h.

STEP 17

The car's speed is **105 km/h**, and the bus's speed is **70 km/h**.

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