a class of 24 students went on a picnic. they had a choice between having a turkey sandwich or a vegetarian sandwich. eighteen students selected turkey, and the rest selected vegetarian. write a ratio to compare the number of turkey sandwiches to the number of vegetarian sandwiches. 4:3 4:1 1:3 3:1 question 8 (5 points) fatima is on the track team. one day at practice, it took her 19 minutes to run 2 miles. select the expression that gives this as a unit rate. $\frac{19 \text{ minutes}}{2 \text{ miles}}$ $9\frac{1}{2}$ miles per minute. $\frac{1 \text{ mile}}{9\frac{1}{2} \text{ minutes}}$

Question

Accepted Answer

### First Question (Ratio of Turkey to Vegetarian Sandwiches)
# Explanation:
## Step1: Find number of vegetarian sandwiches
Total students = 24, Turkey = 18. So Vegetarian = $ 24 - 18 = 6 $.
## Step2: Find the ratio
Ratio of Turkey to Vegetarian = $ 18:6 $. Simplify by dividing both by 6: $ \frac{18}{6}:\frac{6}{6} = 3:1 $.
# Answer: D. 3 : 1

### Second Question (Unit Rate for Fatima's Running)
# Explanation:
## Step1: Recall unit rate definition
Unit rate for time per mile (or mile per minute) – here, to find time per 1 mile, divide total time by total miles.
## Step2: Calculate unit rate
Total time = 19 minutes, total miles = 2. Unit rate (time per mile) = $ \frac{19}{2} = 9\frac{1}{2} $ minutes per mile? Wait, no – wait, the options: Let's re - evaluate. Wait, unit rate for miles per minute: miles divided by time. But the options: The first option is $ \frac{19\ minutes}{2\ miles} $, which is time per mile (unit rate for time per mile). Wait, the third option: $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ is equivalent to $ \frac{2\ miles}{19\ minutes} $ (since 1 mile in $ 9\frac{1}{2}=\frac{19}{2} $ minutes, so 2 miles in 19 minutes). Wait, the correct unit rate for time per mile is $ \frac{19\ minutes}{2\ miles} $, but let's check the options again. Wait, the third option: $ 9\frac{1}{2}=\frac{19}{2} $, so $ \frac{1\ mile}{\frac{19}{2}\ minutes}=\frac{2\ miles}{19\ minutes} $, which is the rate of miles per minute? No, wait, no. Wait, Fatima ran 2 miles in 19 minutes. So miles per minute is $ \frac{2}{19} $ miles per minute, and minutes per mile is $ \frac{19}{2}=9\frac{1}{2} $ minutes per mile. But the options: The first option is $ \frac{19\ minutes}{2\ miles} $ (which is minutes per mile, unit rate for time per mile), the third option is $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ (which is miles per minute? No, $ \frac{1\ mile}{9\frac{1}{2}\ minutes}=\frac{2\ miles}{19\ minutes} $, which is the same as her rate). Wait, the correct unit rate (time per mile) is $ \frac{19}{2}=9\frac{1}{2} $ minutes per mile, but the options: Wait, the first option is $ \frac{19\ minutes}{2\ miles} $, which is the unit rate for time per mile (since it's time divided by distance, giving time per 1 unit of distance). Wait, maybe I misread. Let's check the options again:

Option 1: $ \frac{19\ minutes}{2\ miles} $ – this is time per mile (unit rate for time per mile).

Option 2: $ 9\frac{1}{2}\ miles\ per\ minute $ – that would mean she runs $ 9\frac{1}{2} $ miles in 1 minute, which is wrong (she ran 2 miles in 19 minutes, so way slower).

Option 3: $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ – since $ 9\frac{1}{2}=\frac{19}{2} $, this is equivalent to $ \frac{2\ miles}{19\ minutes} $ (multiply numerator and denominator by 2), which is her actual rate. Wait, but unit rate for miles per minute is $ \frac{2}{19} $ miles per minute, and $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ is the same as $ \frac{2}{19} $ miles per minute (since $ \frac{1}{\frac{19}{2}}=\frac{2}{19} $). But the first option is $ \frac{19\ minutes}{2\ miles} $, which is time per mile (9.5 minutes per mile). Wait, the question says "the expression that gives this as a unit rate". Unit rate can be time per unit distance or distance per unit time. Let's see:

If we want time per 1 mile: $ \frac{19\ minutes}{2\ miles} $ simplifies to $ \frac{9.5\ minutes}{1\ mile} $, which is the unit rate for time per mile.

If we want miles per 1 minute: $ \frac{2\ miles}{19\ minutes}=\frac{1\ mile}{9.5\ minutes} $, which is the third option.

Wait, the question says "the expression that gives this as a unit rate". Let's check the options again. The first option is $ \frac{19\ minutes}{2\ miles} $ (unit rate of minutes per mile), the third option is $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ (unit rate of miles per minute, since it's 1 mile in $ 9\frac{1}{2} $ minutes). Wait, but let's calculate:

To find miles per minute: $ \frac{2\ miles}{19\ minutes}=\frac{1\ mile}{\frac{19}{2}\ minutes}=\frac{1\ mile}{9\frac{1}{2}\ minutes} $, which is the third option.

Wait, but the first option is $ \frac{19\ minutes}{2\ miles} $, which is minutes per mile. Both are unit rates, but let's see the options. Wait, the second option is $ 9\frac{1}{2}\ miles\ per\ minute $, which is wrong because $ \frac{2}{19}\approx0.105 $ miles per minute, not $ 9\frac{1}{2} $. So the correct unit rate (either minutes per mile or miles per minute) – the third option $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ is correct, and the first option $ \frac{19\ minutes}{2\ miles} $ is also a unit rate (minutes per mile). Wait, maybe the question is about time per mile? Wait, no, let's re - read: "Select the expression that gives this as a unit rate."

Fatima ran 2 miles in 19 minutes. Unit rate for miles per minute: $ \frac{2}{19} $ miles per minute, which is not an option. Unit rate for minutes per mile: $ \frac{19}{2}=9\frac{1}{2} $ minutes per mile, which is equivalent to $ \frac{19\ minutes}{2\ miles} $ (first option) or $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ (third option, since $ 9\frac{1}{2}=\frac{19}{2} $, so $ \frac{1\ mile}{\frac{19}{2}\ minutes}=\frac{2\ miles}{19\ minutes} $, which is the original rate, but as a unit rate for 1 mile, it's $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $).

Wait, the first option is $ \frac{19\ minutes}{2\ miles} $ (time per 2 miles? No, unit rate is per 1 unit. So $ \frac{19\ minutes}{2\ miles} $ is the rate for 1 mile (since dividing by 2 gives $ 9\frac{1}{2}\ minutes\ per\ mile $). The third option is $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $, which is the same as $ 1\ mile\ per\ 9\frac{1}{2}\ minutes $, which is a unit rate.

But let's check the options again. The first option is $ \frac{19\ minutes}{2\ miles} $ – this is a rate, but to make it a unit rate (per 1 mile), we can write it as $ \frac{19}{2}\ minutes\ per\ mile = 9\frac{1}{2}\ minutes\ per\ mile $, and the third option is $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $, which is the reciprocal (miles per minute unit rate? No, miles per minute would be $ \frac{2}{19}\approx0.105 $ miles per minute, which is not an option. Wait, I think there's a mistake in my earlier analysis. Wait, the third option: $ 9\frac{1}{2}=\frac{19}{2} $, so $ \frac{1\ mile}{\frac{19}{2}\ minutes}=\frac{2\ miles}{19\ minutes} $, which is the rate at which she ran (2 miles in 19 minutes). So the unit rate for 1 mile is $ \frac{1\ mile}{\frac{19}{2}\ minutes} $, which is the third option. The first option is $ \frac{19\ minutes}{2\ miles} $, which is time per 2 miles, not per 1. Wait, no – unit rate is a rate with a denominator of 1. So to get time per 1 mile, we divide 19 by 2: $ \frac{19}{2}\ minutes\ per\ mile $, which is $ 9\frac{1}{2}\ minutes\ per\ mile $, and the expression for that can be written as $ \frac{19\ minutes}{2\ miles} $ (since when you have 2 miles, you divide by 2 to get 1 mile). Wait, maybe the first option is correct as a unit rate expression (time per mile, with the fraction representing the rate). Alternatively, the third option is also correct. Wait, no, let's calculate:

If it took 19 minutes to run 2 miles, then for 1 mile, time is $ 19\div2 = 9\frac{1}{2} $ minutes. So the unit rate (time per mile) is $ 9\frac{1}{2}\ minutes\ per\ mile $, which can be written as $ \frac{19\ minutes}{2\ miles} $ (since 19 minutes for 2 miles is the same as $ 9\frac{1}{2}\ minutes\ for\ 1\ mile $) or $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ (1 mile in $ 9\frac{1}{2}\ minutes $).

But looking at the options, the first option is $ \frac{19\ minutes}{2\ miles} $, the third is $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $. Wait, the second option is wrong (as $ 9\frac{1}{2}\ miles\ per\ minute $ is way too fast). So between the first and third. Wait, the question says "the expression that gives this as a unit rate". The unit rate for miles per minute would be $ \frac{2}{19}\ miles\ per\ minute $, but that's not an option. The unit rate for minutes per mile is $ \frac{19}{2}\ minutes\ per\ mile $, which is $ \frac{19\ minutes}{2\ miles} $ (first option) or $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ (third option, since $ 9\frac{1}{2}=\frac{19}{2} $, so $ \frac{1\ mile}{\frac{19}{2}\ minutes}=\frac{2\ miles}{19\ minutes} $, which is the original rate, but as a unit rate for 1 mile, it's correct).

Wait, maybe the intended answer is the third option: $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $, because it's a unit rate (1 mile) over time. Or the first option. Wait, let's check the options again:

- Option 1: $ \frac{19\ minutes}{2\ miles} $ – this is a rate, not a unit rate (denominator is 2, not 1). Wait, no – unit rate can be expressed as a fraction where the denominator is the unit (1), but sometimes it's expressed as the rate for the given quantity. Wait, no, unit rate must have a denominator of 1. So to get a unit rate for time per mile, we need to have 1 mile in the denominator. So $ \frac{19\ minutes}{2\ miles}=\frac{9.5\ minutes}{1\ mile} $, which is a unit rate. So the expression $ \frac{19\ minutes}{2\ miles} $ is a way to express the unit rate (since when simplified, it's per 1 mile). Alternatively, $ \frac{1\ mile}{9.5\ minutes} $ is also a unit rate.

But looking at the options, the third option is $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $, which is equivalent to $ \frac{1\ mile}{\frac{19}{2}\ minutes}=\frac{2\ miles}{19\ minutes} $, which is the rate she ran. So the correct unit rate expression (for 1 mile) is $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $.

# Answer: C. $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $ (assuming the third option is C, but based on the given options, the third option is the one with $ \frac{1\ mile}{9\frac{1}{2}\ minutes} $)

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QUESTION IMAGE

Question

Explanation:

First Question (Ratio of Turkey to Vegetarian Sandwiches)

Step1: Find number of vegetarian sandwiches

Step2: Find the ratio

Step1: Recall unit rate definition

Step2: Calculate unit rate

Answer:

Second Question (Unit Rate for Fatima's Running)