The formula to calculate distance is \(\displaystyle d =\frac{r}{t}\) where \(\displaystyle r\) represents rate of speed and \(\displaystyle t\) represents time.

In this problem, \(\displaystyle r = 340\) mph., and \(\displaystyle t= 1\frac{1}{4}\)  hours. 

Convert the \(\displaystyle 1\frac{1}{4}\) to an improper fraction:

\(\displaystyle 1\frac{1}{4} = \frac{(4\times 1)+1}{4} =\frac{5}{4}\)

To solve, multiply.

\(\displaystyle d = \frac{340}{1} \times\frac{5}{4}\)

\(\displaystyle d = \frac{1700}{4}\)

\(\displaystyle d = 425\) miles.

At its cruising speed of \(\displaystyle 340\) mph, the aircraft will travel \(\displaystyle 425\) miles in  \(\displaystyle 1\frac{1}{4}\) hours.

The time is 11:17 AM. In 24 minutes, the minutes will read \(\displaystyle 17 + 24 = 41\), so it wil be 11:41 AM. Let's then add three hours to it.

The time will be 2:41 PM.

He begins with 40 and ends with 4, so 36 cookies total were eaten.  Since he ate 6 himself, that means that his guests ate 30.  Since each guest ate 3 cookies, \(\displaystyle 30/3 =10\) guests.

Each of the children gets 1/4 of the cake. Taking 1/4 of 12 shows that each child gets 3 slices. If Adam finishes 1/3 of his share, then he has eaten one out of the 3 slices. 

To solve this answer, we have to first make the mixed numbers improper fractions so that we can then find a common denominator. To make a mixed number into an improper fraction, you multiply the denominator by the whole number and add the result to the numerator. So, for the presented data:

 \(\displaystyle 1\frac{2}{3}=\frac{(3*1)+2}{3}=\frac{3+2}{3}=\frac{5}{3}\)

and

\(\displaystyle 2\frac{3}{4}=\frac{(4*2)+3}{4}=\frac{8+3}{4}=\frac{11}{4}\)

Now, to find out how many total pounds of vegetables Steven purchased, we need to add these two improper fractions together:

\(\displaystyle \frac{5}{3}+\frac{11}{4}\)  

To add these fractions, they need to have a common denominator. We can adjust each fraction to have a common denominator of \(\displaystyle 12\) by multiplying \(\displaystyle \frac{5}{3}\) by \(\displaystyle \frac{4}{4}\) and \(\displaystyle \frac{11}{4}\)by \(\displaystyle \frac{3}{3}\):

\(\displaystyle (\frac{5}{3}*\frac{4}{4})+(\frac{11}{4}*\frac{3}{3})\)

 To multiply fractions, just multiply across:

\(\displaystyle \frac{20}{12}+\frac{33}{12}\)

We can now add the numerators together; the denominator will stay the same:

\(\displaystyle \frac{53}{12}\)

Since all of the answer choices are mixed numbers, we now need to change our improper fraction answer into a mixed number answer. We can do this by dividing the numerator by the denominator and leaving the remainder as the numerator:

\(\displaystyle 53\div12=4\: r.\:5\)

\(\displaystyle \frac{53}{12}=4\frac{5}{12}\)

This means that our final answer is \(\displaystyle 4\frac{5}{12}\:lbs\).

\(\displaystyle -6\frac{1}{3} + 4\frac{1}{6}\)

To solve, convert all mixed numbers to improper fractions:

\(\displaystyle -6\frac{1}{3} = -\frac{(3\times 6) +1}{3} = -\frac{19}{3}\)

\(\displaystyle 4\frac{1}{6} = \frac{(6\times 4)+1}{6} = \frac{25}{6}\)

Because the denominators are not the same you have to find the Least Common Multiple, which is the Least Common Denominator of 3 and 6, which is 6.  Then convert both fractions, if needed, to equivalent fractions with 6 as the denominator.

\(\displaystyle -\frac{19}{3} =-\frac{38}{6}\) These are equivalent because the numerator and the denominator have been multiplied by 2 to get the equivalent fraction.

\(\displaystyle -\frac{38}{6} +\frac{25}{6} = -\frac{13}{6} = -2\frac{1}{6}\) 

The answer is negative because the numerator with the greatest absolute value is -38 and that is a negative number.

\(\displaystyle -\frac{1}{3} - (-\frac{3}{4}) =\)

\(\displaystyle -\frac{1}{3} +\frac{3}{4}\)  The subtraction of a negative changes the subtraction sign to a positive.

Find the LCD of 3 and 4 and convert to equivalent fractions, with the LCD as the denominator for both fractions.

The LCD is 12.

\(\displaystyle -\frac{1}{3} = -\frac{4}{12}\)   The numerator and the denominator were multiplied by 4.

\(\displaystyle \frac{3}{4} = \frac{9}{12}\)  The numerator and the denominator were multiplied by 3.

\(\displaystyle -\frac{4}{12} + \frac{9}{12} = \frac{5}{12}\)

Because the absolute value of 9 is greater than the absolute value of -4, the answer is positive.

Evaluate   \(\displaystyle x-y\)    if    \(\displaystyle x= -\frac{5}{8}\)    and    \(\displaystyle y = 2\frac{5}{6}\).

 \(\displaystyle -\frac{5}{8} -2\frac{5}{6}\)

First, convert any mixed numbers to improper fractions.

\(\displaystyle 2\frac{5}{6} = \frac{(6\times 2)+5}{6} = \frac{17}{6}\)

\(\displaystyle -\frac{5}{8} - \frac{17}{6}\) 

In order to solve, find the common denominator of 8 and 6. This is found by listing the multiples and picking the LCM or the least common multiple.  This becomes the LCD or Least Common Denominator.

Multiples of 8 are 8, 16, 24.

Multiples of 6 are 6, 12,18, and 24.

The LCM or the LCD will be 24.

Create equivalent fractions for both with 24 as the denominator,

\(\displaystyle -\frac{5}{8} =-\frac{15}{24}\)    Both numerator and denominator were multiplied by 3.

\(\displaystyle -\frac{17}{6} = -\frac{68}{24}\)  Both numerator and denominator were multiplied by 4.

\(\displaystyle -\frac{15}{24} + (-\frac{68}{24}) = \frac{-15 + (-68)}{24} = -\frac{83}{24}\)  When adding two negatives, the result is negative.

\(\displaystyle -\frac{83}{24} = -3\frac{11}{24}\)

\(\displaystyle 14 -3\frac{1}{8}\)

Change all whole numbers and mixed numbers to improper fractions.

\(\displaystyle 14 = \frac{14}{1}\)  When there is a whole number, just place the number 1 underneath the fraction bar as the denominator.

\(\displaystyle 3\frac{1}{8} = \frac{(8\times 3)+1}{8} = \frac{25}{8}\)

Because the denominators are different, create equivalent fractions by using the Least Common Denominator.  The LCD of 8 and 1 is 8.

\(\displaystyle \frac{14}{1} =\frac{112}{8}\)

\(\displaystyle \frac{25}{8}\)    does not have to be changed, because the denominator is already the              LCD, which is 8.

Now that the denominators are the same, subtract.

\(\displaystyle \frac{112}{8} -\frac{25}{8} = \frac{112-25}{8} = \frac{87}{8}\)

\(\displaystyle \frac{87}{8} = 10\frac{7}{8}\)

Evaluate \(\displaystyle a-b\) if    \(\displaystyle a = \frac{5}{7}\)     and   \(\displaystyle b = -\frac{2}{5}\)

\(\displaystyle \frac{5}{7} - (-\frac{2}{5}) =\)

\(\displaystyle \frac{5}{7} +\frac{2}{5}\)  The subtraction of a negative changes the operation from subtraction to addition.

Find the Least Common Multiple of 7 and 5.

Multiples of 7 are 7, 14, 21, 28, 35.

Multiples of 5 are 5, 10, 15, 20 ,25, and 35.

You can also just multiply the denominators.

The LCM  or the LCD of 7 and 5 is 35. 

Create equivalent fractions with 35 as the denominator.

\(\displaystyle \frac{5}{7} =\frac{25}{35}\)   Both the numerator and denominator were multiplied by 5.

\(\displaystyle \frac{2}{5} = \frac{14}{35}\)    Both the numerator and denominator were multiplied by 7.

Now that the denominators are the same, just add the numerators.

 \(\displaystyle \frac{25}{35} +\frac{14}{35} = \frac{25+14}{35} = \frac{39}{35 } = 1\frac{4}{35}\)