Question

an object is traveling at a steady speed of $10\frac{1}{10}$ km/h. how long will it take the object to trav nearest integer to find the estimated answer. then find the exact answer.
estimate the answer.
it takes the object about 12 minute(s) to travel $1\frac{9}{10}$ km.
find the exact answer.
it takes the object \\(\square\\) minute(s) to travel $1\frac{9}{10}$ km.
(simplify your answer. type an integer, proper fraction, or mixed number.)

Explanation:

Step1: Recall the formula for time

The formula for time \( t \) is \( t=\frac{d}{v} \), where \( d \) is distance and \( v \) is speed. First, convert the speed and distance to improper fractions. The speed \( v = 10\frac{1}{10}=\frac{10\times10 + 1}{10}=\frac{101}{10}\) km/h, and the distance \( d = 1\frac{9}{10}=\frac{1\times10+9}{10}=\frac{19}{10}\) km.

Step2: Calculate time in hours

Using the formula \( t=\frac{d}{v} \), we substitute the values: \( t=\frac{\frac{19}{10}}{\frac{101}{10}}=\frac{19}{10}\times\frac{10}{101}=\frac{19}{101}\) hours.

Step3: Convert hours to minutes

Since 1 hour = 60 minutes, we multiply the time in hours by 60: \( t=\frac{19}{101}\times60=\frac{1140}{101}\approx11.287\), but we need the exact value. Wait, let's check the calculation again. Wait, maybe I made a mistake in the speed. Wait, the speed is \( 10\frac{1}{10} \) km per hour, which is \( \frac{101}{10} \) km/h, and distance is \( 1\frac{9}{10}=\frac{19}{10} \) km. So time in hours is \( \frac{19/10}{101/10}=\frac{19}{101} \) hours. Then convert to minutes: \( \frac{19}{101}\times60=\frac{1140}{101}=11\frac{29}{101} \)? Wait, no, wait, maybe the problem is that the speed is \( 10\frac{1}{10} \) km per hour, but maybe the distance is \( 1\frac{9}{10} \) km. Wait, let's re - do the calculation.

Wait, the formula for time is \( t=\frac{d}{v} \). Let's write all values as fractions:

\( v = 10\frac{1}{10}=\frac{101}{10} \) km/h, \( d = 1\frac{9}{10}=\frac{19}{10} \) km.

\( t=\frac{d}{v}=\frac{\frac{19}{10}}{\frac{101}{10}}=\frac{19}{101} \) hours.

To convert hours to minutes, multiply by 60: \( t=\frac{19}{101}\times60=\frac{1140}{101}=11\frac{29}{101} \)? Wait, that can't be right. Wait, maybe I misread the speed. Wait, maybe the speed is \( 10\frac{1}{10} \) km per hour, but maybe the distance is \( 1\frac{9}{10} \) km. Wait, let's check the problem again. The problem says "An object is traveling at a steady speed of \( 10\frac{1}{10} \) km/h. How long will it take the object to travel \( 1\frac{9}{10} \) km? Use the nearest integer to find the estimated answer. Then find the exact answer."

Wait, maybe I made a mistake in the formula. Time = Distance / Speed. So Distance is \( 1\frac{9}{10}=\frac{19}{10} \) km, Speed is \( 10\frac{1}{10}=\frac{101}{10} \) km/h. So Time (in hours) is \( \frac{19/10}{101/10}=\frac{19}{101} \) hours. Then convert to minutes: \( \frac{19}{101}\times60=\frac{1140}{101}=11\frac{29}{101} \approx 11.29\). But the estimated answer was 12, but the exact answer is \( \frac{1140}{101} \) or simplified. Wait, no, maybe the speed is \( 10.1 \) km/h and distance is \( 1.9 \) km. Then time in hours is \( 1.9\div10.1\approx0.188 \) hours. Convert to minutes: \( 0.188\times60\approx11.28 \) minutes. So the exact answer is \( \frac{1.9}{10.1}\times60=\frac{19}{101}\times60=\frac{1140}{101}=11\frac{29}{101} \) minutes? Wait, no, let's do the fraction division correctly.

\( t=\frac{d}{v}=\frac{1\frac{9}{10}}{10\frac{1}{10}}=\frac{\frac{19}{10}}{\frac{101}{10}}=\frac{19}{101} \) hours. Multiply by 60: \( \frac{19\times60}{101}=\frac{1140}{101}=11\frac{29}{101} \) minutes. Wait, but maybe the problem has a typo, or maybe I misread the speed. Wait, if the speed was \( 10\frac{1}{10} \) km per hour, and distance is \( 1\frac{9}{10} \) km, then the exact time in minutes is \( \frac{19}{101}\times60=\frac{1140}{101}=11\frac{29}{101} \approx 11.29 \) minutes. But let's check the calculation again.

Wait, another way: Let's use decimals. Speed \( v = 10.1 \) km/h, distance \( d = 1.9 \) km. Time \( t=\frac{d}{v}=\frac{1.9}{10.1}\) hours. Convert to minutes: \( \frac{1.9}{10.1}\times60=\frac{114}{10.1}=\frac{1140}{101}=11\frac{29}{101} \) minutes. So the exact answer is \( \frac{1140}{101} \) or \( 11\frac{29}{101} \) minutes.

Wait, but maybe I made a mistake in the speed. Wait, the problem says "10 1/10 km/h" which is 10.1 km/h, and distance is "1 9/10 km" which is 1.9 km. So time in hours is 1.9 / 10.1, then times 60. So 1.9*60 = 114, 114 / 10.1 = 1140 / 101 = 11.287... So the exact answer is 1140/101, which simplifies to 11 29/101 minutes.

Answer:

\( \frac{1140}{101} \) (or \( 11\frac{29}{101} \))