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# 4.2: Real Numbers Overview

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## Arithmetic of Signed Numbers

### Real Numbers Overview

(0,35)(0,-35) (0,0)(1,0)40 (0,0)(0,-1)30 (40,-30)(-1,0)40 (40,-30)(0,1)30 (17,0.5)Real Numbers (1,-1.5)Irrational Numbers like $$\sqrt{2}$$, $$-\pi$$ (5,-5)(1,0)30 (5,-5)(0,-1)20 (35,-25)(-1,0)30 (35,-25)(0,1)20 (7,-7)Rational Numbers like $$\displaystyle \frac{-23}{7}$$, $$\displaystyle \frac{7}{1}$$, $$\displaystyle \frac{0}{1}$$ (10,-10)(1,0)20 (10,-10)(0,-1)10 (30,-20)(-1,0)20 (30,-20)(0,1)10 (12,-12)Integers like $$-3$$, $$7$$, $$0$$ (15,-13)(1,0)12.5 (15,-13)(0,-1)5.3 (27.5,-18.5)(-1,0)12.5 (27.5,-18.3)(0,1)5.4 (15,-14.5)Whole Numbers $$0$$, $$1$$, $$2$$ (16,-15)(1,0)9 (16,-15)(0,-1)2.5 (25,-17.5)(-1,0)9 (25,-17.5)(0,1)2.5 (16.2,-16.0)Natural Numbers (16.4,-17.0)$$1$$, $$2$$ (12.4,-33.0)Also known as Counting Numbers (12.4,-34.5)or as Positive Integers (20,-32)(0,1)15 (20.4,-31.5)Also known as Non-Negative Integers (31.5,-30.5)(-1,2)6.3

© H. Feiner 2011, 02015, 2019

(0,15)(0,-3) (0,-2.5)(1,0)35 (1,-2.8)(2,0)17(0,1)1 (0.5,-1.3)$$-8$$ (2.5,-1.3)$$-7$$ (4.5,-1.3)$$-6$$ (6.5,-1.3)$$-5$$ (8.5,-1.3)$$-4$$ (10.5,-1.3)$$-3$$ (12.5,-1.3)$$-2$$ (14.5,-1.3)$$-1$$ (16.7,-1.3)$$0$$ (18.7,-1.3)$$1$$ (20.7,-1.3)$$2$$ (22.7,-1.3)$$3$$ (24.7,-1.3)$$4$$ (26.7,-1.3)$$5$$ (28.7,-1.3)$$6$$ (30.7,-1.3)$$7$$ (32.7,-1.3)$$8$$ (19.0,1) (21.0,1) (23.0,1) (25.0,1) (27.0,1) (29.0,1) (31.0,1) (33.0,1) (19.0,1)(1,0)15.0 (34.5,1)Natural or Counting Numbers (17.0,3) (19.0,3) (21.0,3) (23.0,3) (25.0,3) (27.0,3) (29.0,3) (31.0,3) (33.0,3) (17.0,3)(1,0)17.0 (34.5,3)Whole Numbers (0.5,5) (2.5,5) (4.5,5) (6.5,5) (8.5,5) (10.5,5) (12.5,5) (14.5,5) (16.7,5) (18.7,5) (20.7,5) (22.7,5) (24.7,5) (26.7,5) (28.7,5) (30.7,5) (32.7,5) (0,5)(1,0)34 (34.5,5)Integers (1,7)(0.61,0)53 (0,7)(1,0)34 (34.5,7)Rational Numbers (1,9)(0.61,0)53 (0,9)(1,0)34 (34.5,9)Irrational Numbers

Why does the number of rational and the number of irrational numbers appear to be the same on the number line above?

A rational number (ratio of two integers) can be rewritten as a terminating decimal (like $$\displaystyle \frac{5}{2}=2.5$$) or a number with infinitely many decimals in repeating groups (like $$\displaystyle \frac{5}{11}=0.45454545454545\cdots$$)

An irrational number can be represented as a decimal number with infinitely many decimals, but no group of digits repeats itself.

Can we construct an irrational number?

Start with a decimal, like $$0.2$$. Append a different group of digits, like $$12$$ to get $$0.212$$. Then append a different group, like $$112$$ to get $$0.212112$$. Keep appending groups of digits each different from the previous ones by an additional digit 1. This guarantees that there are no groups of repeating decimals. The number is $$0.212112111211112111112\cdots$$

*********************************************************

One method is to think of the balance in your checking account.

Example: $$5+2=7$$. You start with $$$5$$ in your checking account and deposit$$$2$$. You now have $$$7$$ in your account. Example: $$5+(-2)=3$$. You start with$$$5$$ in your checking account and write a check for $$$2$$. You now have$$$3$$ in your account. (The purpose of the parentheses is to separate $$+$$ and $$-$$)
Example: $$-5+2=-3$$. You start with $$$-5$$ in your checking account (you are overdrawn) and deposit$$$2$$. You now have $$$-3$$ in your account (you are still overdrawn$$$3$$).
Example: $$-5+(-2)=-7$$. You start with $$$-5$$ in your checking account (you are overdrawn) and write a check for$$$2$$. You now have $$$-7$$ in your account (you are overdrawn$$$7$$).
A second method is to use arrows.

A positive number is represented by an arrow pointing in the positive direction (to the right). The length of the arrow is the number of units – tick marks on a number line – indicated by the number (plus one).

(0,2)(0,-1) (17,0.5)(1,0)10.0 (0,0.5)The number $$5$$ is represented by (17,0)(2,0)6(0,1)1 (16.7,-1.3)$$0$$ (18.7,-1.3)$$1$$ (20.7,-1.3)$$2$$ (22.7,-1.3)$$3$$ (24.7,-1.3)$$4$$ (26.7,-1.3)$$5$$

A negative number is represented by an arrow pointing in the negative direction (to the left). The length of the arrow is the number of units – tick marks on a number line – indicated by the number (plus one).

(0,1)(0,0) (21,0.5)(-1,0)4.0 (0,0.5)The number $$-2$$ is represented by (17,0)(2,0)3(0,1)1 (16.7,-1.3)$$-2$$ (18.7,-1.3)$$-1$$ (20.7,-1.3)$$0$$

Example: $$5+2=7$$.

(0,5)(0,-3) (0,0)(1,0)35 (1,-0.5)(2,0)17(0,1)1 (0.5,-1.3)$$-8$$ (2.5,-1.3)$$-7$$ (4.5,-1.3)$$-6$$ (6.5,-1.3)$$-5$$ (8.5,-1.3)$$-4$$ (10.5,-1.3)$$-3$$ (12.5,-1.3)$$-2$$ (14.5,-1.3)$$-1$$ (16.7,-1.3)$$0$$ (18.7,-1.3)$$1$$ (20.7,-1.3)$$2$$ (22.7,-1.3)$$3$$ (24.7,-1.3)$$4$$ (26.7,-1.3)$$5$$ (28.7,-1.3)$$6$$ (30.7,-1.3)$$7$$ (32.7,-1.3)$$8$$ (31.0,0) (21.0,1.5)$$5$$ (26.5,1.5)$$+$$ (29.0,1.35)$$2$$ (27.0,1)(1,0)3.5 (17.5,1)(1,0)9.5

Example: $$5+(-2)=3$$.

(0,5)(0,-3) (0,0)(1,0)35 (1,-0.5)(2,0)17(0,1)1 (0.5,-1.3)$$-8$$ (2.5,-1.3)$$-7$$ (4.5,-1.3)$$-6$$ (6.5,-1.3)$$-5$$ (8.5,-1.3)$$-4$$ (10.5,-1.3)$$-3$$ (12.5,-1.3)$$-2$$ (14.5,-1.3)$$-1$$ (16.7,-1.3)$$0$$ (18.7,-1.3)$$1$$ (20.7,-1.3)$$2$$ (22.7,-1.3)$$3$$ (24.7,-1.3)$$4$$ (26.7,-1.3)$$5$$ (28.7,-1.3)$$6$$ (30.7,-1.3)$$7$$ (32.7,-1.3)$$8$$ (23.0,0) (19.0,1.5)$$5$$ (24.0,2.5)$$-2$$ (27.0,2)(-1,0)4.5 (17.5,1)(1,0)9.5

Example: $$-5+2=-3$$.

(0,5)(0,-3) (0,0)(1,0)35 (1,-0.5)(2,0)17(0,1)1 (0.5,-1.3)$$-8$$ (2.5,-1.3)$$-7$$ (4.5,-1.3)$$-6$$ (6.5,-1.3)$$-5$$ (8.5,-1.3)$$-4$$ (10.5,-1.3)$$-3$$ (12.5,-1.3)$$-2$$ (14.5,-1.3)$$-1$$ (16.7,-1.3)$$0$$ (18.7,-1.3)$$1$$ (20.7,-1.3)$$2$$ (22.7,-1.3)$$3$$ (24.7,-1.3)$$4$$ (26.7,-1.3)$$5$$ (28.7,-1.3)$$6$$ (30.7,-1.3)$$7$$ (32.7,-1.3)$$8$$ (11.0,0) (15.0,1.5)$$-5$$ (7.0,2.5)$$+\hspace{0.2in}2$$ (7.0,2)(1,0)4.5 (17.0,1)(-1,0)10.0

Example: $$-5+(-2)=-7$$.

(0,5.5)(0,-3) (0,0)(1,0)35 (1,-0.5)(2,0)17(0,1)1 (0.5,-1.3)$$-8$$ (2.5,-1.3)$$-7$$ (4.5,-1.3)$$-6$$ (6.5,-1.3)$$-5$$ (8.5,-1.3)$$-4$$ (10.5,-1.3)$$-3$$ (12.5,-1.3)$$-2$$ (14.5,-1.3)$$-1$$ (16.7,-1.3)$$0$$ (18.7,-1.3)$$1$$ (20.7,-1.3)$$2$$ (22.7,-1.3)$$3$$ (24.7,-1.3)$$4$$ (26.7,-1.3)$$5$$ (28.7,-1.3)$$6$$ (30.7,-1.3)$$7$$ (32.7,-1.3)$$8$$ (3.0,0) (13.0,1.5)$$-5$$ (4.0,1.5)$$-2$$ (7.0,1.5)$$+$$ (7.0,1)(-1,0)4.0 (17.0,1)(-1,0)10.0

Still another method is to follow a set of rules.
To add two numbers of the same (positive) sign like $$5+2$$:

Obtain the absolute value of each number, $$|5|=5$$ and $$|2|=2$$.

Add the absolute values $$5+2=7$$.

The sign of the sum is the sign of the original numbers, namely positive.
Another example:

To Add two numbers of the same (negative) sign like $$(-5)+(-2)$$:

Obtain the absolute value of each number, $$|-5|=5$$ and $$|-2|=2$$.

Add the absolute values $$5+2=7$$.

The sign of the sum is the sign of the original numbers, namely negative. $$(-5)+(-2)=-7$$
A further example:

To Add two numbers of opposite signs like $$(-5)+2$$:

Obtain the absolute value of each number, $$|-5|=5$$ and $$|2|=2$$.

Subtract the smaller absolute values from the larger one. $$|-5|-|2|=3$$.

The sign of the sum is negative in this example. It is the sign of the larger absolute value ($$|-5|$$) before supplying absolute value bars. $$(-5)+2=-3$$
Here is one last method involving mathematical manipulatives suggested by the University of Utah
( nlvm.usu.edu/en/nav/frames$$\_$$asid$$\_$$161$$\_$$g$$\_$$2$$\_$$t$$\_$$1.html?from=search.html)
$$5+2\Rightarrow (+++++)\hbox{ added to }(++)=+++++++=7$$
$$5+(-2)\Rightarrow (+++++)\hbox{ addea to } (--)=(\pm)(\pm)+++=3$$
Each $$+-$$ or $$\pm$$ eliminate each other resulting in $$0$$.
$$-5+2\Rightarrow (-----)\hbox{ added to } (++)=(\pm \pm -)---=-3$$
$$-5+(-2)\Rightarrow (-----)\hbox{ added to } (--)=-------=-7$$

### Subtraction of Signed Numbers

Treat subtraction of signed numbers as addition of signed numbers by making two changes:

1) change minus to plus (subtraction to addition.)

2) Add the opposite of the number you originally subtracted.
Exampls:

$$5-2=5+(-2)=3$$
$$5-(-2)=5+(+2)=7$$
$$-5-2=-5+(-2)=-7$$
$$-5-(-2)=-5+(2)=-3$$
Be careful about the meaning of “difference" and “subtract from".

The difference of $$9$$ and $$5$$ is $$9-5=4$$,

but

subtract $$9$$ from $$5$$ is $$5-9=-4$$.

### Multiplication of Signed Numbers

$$4(-3)=(-3)+(-3)+(-3)+(-3)=-12$$

We can use this reasoning whenever we multiply two integers of different signs. Thus the product of a positive number and a negative number is negative.

Rewrite $$4(-3)$$ as $$-[4(3)]$$, that is isolate the negative sign and carry out the multiplication of what is left from the product.

(0,7)(0,-6) (0,-3)$$(-\underbrace{4)(-3)}=-[4(-3)]=-[-12]=12$$ (0,0)Isolate the negative sign. (1,-0.5)(0,-1)2 (8,-0.5)(0,-1)2 (12.5,-1.0)Product of a positive and a negative number. (13.0,-0.5)(-1,-1)1.5 (17,-7)The opposite of $$-12$$. (21.0,-6.0)(1,4)0.7

$$(-3)^2=(-3)(-3)=9$$

but

$$-3^2=-(3)(3)=-9$$. Be careful. Lots of students use exponentiation of signed numbers incorrectly.

Those who claim that $$-5^2=25$$ belong to the IPS (Invisible Parentheses Society). If you are a member of the IPS, give up your membership immediately.

Warning:

Do not say “negative AND negative is positive." AND s usually taken to mean addition. The sum of two negative numbers is negative.

Say instead “The product (or quotient for division) of two negative numbers is positive."

### Division of Signed Numbers

Division is repeated subtraction.

(0,12)(-1,-12) (-1.0,-1.3)$$\displaystyle \frac{1\ 2}{4}=1\ 2 \div \ 4\Rightarrow$$ (8.5,-2.3)$$-4$$ 1 (8.5,-1.3)$$1\ 2\ \ \ \$$ (10.5,-2.05) (8.2,-0.3)(1,0)2.0 (8.2,-0.3)(0,-1)1.2 (8.5,-2.6)(1,0)1.7 (9.3,-3.8)$$8$$ (8.5,-5.2)$$-4$$ 2 (10.5,-5.0) (8.2,-5.6)(1,0)1.7 (9.3,-6.8)$$4$$ (8.5,-8.2)$$-4$$ 3 (10.5,-8.0) (8.2,-8.6)(1,0)1.7 (9.3,-9.8)$$0$$ (8.5,-10.8)$$-4$$ Cannot be negative (8.5,-11.8) Stop subtracting. (11.5,-8.0)Subtract 4 from 12 (11.5,-9.0)exactly 3 times.

Here is a more useful definition of division in terms of multiplication. Convert division to multiplication. $$12\div 4=3$$ or $$4 \cdot 3=12$$.

(0,10)(-2,-7) (0.5,-1.3)$$1\ 2 \div \ 4=\displaystyle \frac{1\ 2}{4}=\ 3$$ then $$\underbrace{4 \ \times \ 3}\ =\ 12$$ (6.55,-1.8) (9.7,-1.0) (8.5,-1.8)(-3,-1)1.0 (8.5,-1.8)(3,1)1.0 (8.5,-2.1)(1,0)6.8 (6.6,0.5)(0,1)1.5 (6.6,2.0)(1,0)13.3 (20.0,2.0)(0,-1)2.5

Examples:

$$12\div (-3)=\displaystyle \frac{12}{-3}=-4$$ because $$(-3)(-4)=12$$
$$-12\div (3)=\displaystyle \frac{-12}{3}=-4$$ because $$(3)(-4)=-12$$
$$-12\div (-3)=\displaystyle \frac{-12}{-3}=4$$ because $$(-3)(4)=-12$$
The sign rules of a quotient are identical to the product sign rules:

The quotient of two same signed numbers is positive.

The quotient of two opposite signed numbers is negative.

Division involving 0:

$$0\div (3)=\displaystyle \frac{0}{3}=0$$ because $$(\hbox{dividend }0)(\hbox{divisor }3)$$ $$=\hbox{quotient }0$$

$$12\div (0)=\displaystyle \frac{12}{0}=x$$ (an unknown number in the $$x$$-box).
$$x$$ is undefined because $$(0)(x)=0\not = 12$$.

The operation of division by $$0$$ is undefined. It is incorrect to say that $$0$$ is undefined (unless you need to explain your grades to your parents or spouse.)

### Evaluation:

Example:

Evaluate $$\displaystyle \frac{2x-4(y+3)}{3x-y+9}$$ if $$x=-3$$ and $$y=-2$$.

Solution;

$$\begin{array}{rcl lll} \!\!\!\!\!\displaystyle \frac{2x-4(y+3)}{3x-y+9}\!\!&=&\displaystyle \frac{2()-4[()+3]}{3()-()-9}\ \ \ \hbox{Replace variables by parentheses.}\\[15pt] &=&\displaystyle \frac{2(-3)-4[(-2)+3]}{3(-3)-(-2)+9}\ \ \ \hbox{Drop values of variables into ( ).}\\[15pt] &=&\displaystyle \frac{-6-4[1]}{-9-(-2)+9}&\hbox{}\\[15pt] &=&\displaystyle \frac{-6-4}{-9+(2)+9}\\[10pt] &=&\displaystyle \frac{-10}{2}=-5\\[10pt] %&=&\displaystyle \frac{-10}{2} \end{array}$$
Exercise 4

1. Name a whole number that is not a counting number.

2. Why is every integer a rational number?

3. Which rational number does not have a reciprocal?

4. There are infinitely many irrational numbers between any two rational numbers. Construct an irrational number between $$\displaystyle \frac{1}{5}$$ and $$\displaystyle \frac{1}{6}$$.

5. 
$$\!\!\!\begin{array}{|r|r|r|r|rr| rr|r| rrr| r|r|r|}\hline &&& \\[-3pt] 12+(-7)=&-11+9=&-14+(-7)=&10-(-7)=\\$$ $$&&& \\[-3pt] \hline &&& \\[-3pt] -16-(-25)=&-11-19=&-8-(-5)=&20-(-7)=\\ &&& \\[-1pt] \hline %&&& &&& &&& &&\\[-7pt] \end{array}$$

6. 
$$\!\!\!\!\!\begin{array}{|r|r|r| r|rr| rrr| rrr| r|r|r|}\hline &&& \\[-3pt] 14(-2)=&-11\cdot 9=&-14\cdot (-7)=&70\div (-7)=\\$$ $$&&& \\[-3pt] \hline &&& \\[-3pt] \!\!\!-100\!\div\![-(-25)]\!=\!&\!\!-19\!\div\! (-38)\!=\!&\!\!-[-8\!\div\! (-40)]\!=\!&20\!\div \!(-85)\!=\!\\ &&& \\[-1pt] \hline %&&& &&& &&& &&\\[-7pt] \end{array}$$

7. What number is subtracted from $$-10$$ to get a difference of $$30$$?

8. Mount McKinley in Alaska is at $$20,300$$ ft above sea level and Death Valley in California is at $$280$$ ft below sea level. Draw a vertical number line to identify altitude, with positive upward. Place a dot at sea level, at Death Valley and at Mount McKinley. Label each point with a signed number. What is the distance (in ft) between Mount McKinley and Death Valley?

9. The temperature on a cold Winter day in Luxembourg was measured four times. Three measurements were $$4^{\circ}$$ C, $$-7^{\circ}$$ C, and $$8^{\circ}$$ C. The average was $$-1^{\circ}$$ C. The fourth measurement was lost. Find the lost measurement if the average for that day was $$-1^{\circ}$$ C

10. A decorator from Exquisite Affairs blows helium balloons for a birthday celebration. Each balloon is tied to a $$9$$ ft string. The decorator has $$256$$ yds of string on a roll. Does the decorator have enough string to blow $$100$$ balloons? If yes, is there any string left? If no, how many balloons can be blown? There are $$3$$ feet per yard.

11. A restaurant cook has $$25$$ kilograms (kg) of potatoes. Each order on a banquet menu requires $$3$$ potatoes on the average. Assume the average weight of a potato to be $$100$$ grams (g). There are $$27$$ people attending the banquet. Does the cook have enough potatoes to serve the banquet? There are $$1,000$$ g in a kg.

The solutions follow:

Keep them covered up till you have worked out each of the problems above without help from any source.

1. Name a whole number that is not a counting number.
Solution:
The set (collection) of whole numbers is $$\{0,1,2,3,\cdots\}$$ All of these numbers are alo counting numbers except for $$0$$. Thus $$0$$ is a whole number (the only whole number) that is not a counting number.

2. Why is every integer a rational number?
Solution:
Every integer, like 6, can be written as a rational number (ratio, fraction) whose denominator is 1. $$6=\displaystyle \frac{6}{1}$$.

3. Which rational number does not have a reciprocal?
Solution:
[10pt] $$0$$ does not have a reciprocal because $$\displaystyle \frac{1}{0}$$ does not exist.

4. There are infinitely many irrational numbers between any two rational numbers. Construct an irrational number between $$\displaystyle \frac{1}{5}$$ and $$\displaystyle \frac{1}{6}$$.
Solution:
[10pt] $$\displaystyle \frac{1}{5}=0.2$$
$$\displaystyle \frac{1}{6}=0.1666\cdots$$.
The number $$N$$ to be constructed must not have any group of repeating decimals, as follows (the answer is not unique):
$$N=0.17\ 117\ 1117\ 11117\ \cdots$$

5. Solution:
$$\begin{array}{rcr ||rcr rcl rcl rrr} &&& &&& &&& &&\\[10pt] 12+(-7)&=&5,& -11+9&=&-2, \\[10pt] -14+(-7)&=&-21, &10-(-7)&=&17,\\[10pt]%   &&& &&& &&& && %\\[-3pt] \hline &&& &&& &&& &&\\[10pt] -16-(-25)&=&9, &-11-19&=&-30, \\[10pt] -8-(-5)&=&-3, &20-(-7)&=&27,\\ % &&& &&& &&& &&\\[10pt] %\hline %&&& &&& &&& &&\\[-7pt] \end{array}$$

6. Solution:
$$\begin{array}{rcr ||rcr rcl rcl rrr} 14(-2)&=&-28, &-11\cdot 9&=&-99\\[10pt] -14\cdot (-7)&=&98, &70\div (-7)&=&-10\\[10pt]$$ $$\\[-15pt]\\[-12pt] -100\div[-(-25)]&=&-4, &-19\div (-38)&=&\displaystyle \frac{1}{2}, \\[13pt] -[-8\div (-40)]&=&\displaystyle -\frac{1}{5}, &20\div (-85)&=&\displaystyle -\frac{4}{17}\\ \end{array}$$

7. $$\!\!$$What number is subtracted from $$\!-10$$ for a difference of $$30$$?
Solution:
[10pt] Let $$x$$ be the number. Then $$-10-x=30$$. $$x$$ has to be negative because subtracting a negative number is equivalent to adding a positive number. Then $$x=-40$$ because $$-10-(-40)=30\ \Rightarrow \ -10+40=30$$.

8. Mount McKinley in Alaska is at $$20,300$$ ft above sea level and Death Valley in California is at $$280$$ ft below sea level. Draw a vertical number line to identify altitude, with positive upward. Place a dot at sea level, at Death Valley and at Mount McKinley. Label each point with a signed number. What is the distance (in ft) between Mount McKinley and Death Valley?
Solution:
[10pt]

(0,7)(3,-6) (10.0,-7)(0,1)8 (10.0,-6) (1.0,-6)Death Valley (11.0,-6)$$-280$$ (10.0,-4) (1.0,-4)sea level (11.0,-4)0 (10.0,0) (1.0,0)Mount McKinley (11.0,0)$$20,300$$ (15.0,-2)The distance between Mount (15.0,-3)McKinley and Death Valley is (14.0,-5) $$20,300\!-\!(-280)\!=\!20,300\!+\!280\!=20,580\!$$ ft.

9. The temperature on a cold Winter day in Luxembourg was measured four times. Three measurements were $$4^{\circ}$$ C, $$-7^{\circ}$$ C, and $$8^{\circ}$$ C. The average was $$-1^{\circ}$$ C. The fourth measurement was lost. Find the lost measurement if the average for that day was $$-1^{\circ}$$ C
Solution:
[10pt] Since the average of the four temperatures is $$-1$$, the sum of the four temperatures is $$(4)(-1)=-4$$. The sum of the three recorded temperatures is $$(4)+(-7)+(8)=-3+8=5$$. The lost temperature is $$-4-5=-9^{\circ}$$ C.

10. A decorator from Exquisite Affairs blows helium balloons for a birthday celebration. Each balloon is tied to a $$9$$ ft string. The decorator has $$256$$ yds of string on a roll. Does the decorator have enough string to blow $$100$$ balloons? If yes, is there any string left? If no, how many balloons can be blown? There are $$3$$ feet per yard.
Solution:
$$\displaystyle \frac{256\hbox{ yds}}{1}\cdot \frac{3\hbox{ ft}}{\hbox{ yd}}=768$$ ft.
The first balloon requires $$9$$ ft of string. The decorator has $$768-9=759$$ ft left.
The second balloon requires $$9$$ ft of string. The decorator has $$759-9=750$$ ft left.
This repeated subtraction of $$9$$ ft implies division.
The number of balloons that can be blown is
$$\displaystyle \frac{768\hbox{ ft}}{1}\cdot \frac{1 \hbox{ balloon}}{9\hbox{ ft}}=85\frac{1}{3}$$ balloons.
The decorator can blow (and tie) $$85$$ balloons. She needs to tie $$100-85=15$$ more balloons. She is short $$15\cdot 9$$ ft $$=135$$ more ft of string. (The $$3$$ ft of sting left from the roll of string is wasted.)

11. A restaurant cook has $$25$$ kilograms (kg) of potatoes. Each order on a banquet menu requires 3 potatoes on the average. Assume the average weight of a potato to be $$100$$ grams (g). There are $$27$$ people attending the banquet. Does the cook have enough potatoes to serve the banquet? There are $$1,000$$ g in a kg.
Solution:
The cook needs to fill $$27$$ orders.
The number of potatoes needed is
$$\displaystyle \frac{27 \hbox{ orders}}{1}\cdot \frac{3 \hbox{ potatoes}}{1 \hbox{ order}}=81$$ potatoes.
The $$81$$ potatoes weigh $$\displaystyle \frac{81 \hbox{ potatoes}}{1}\cdot \frac{100\hbox{ g}}{1\hbox{ potato}}=8,100$$ g.
The number of kg needed is $$\displaystyle \frac{8,100\hbox{ g}}{1}\cdot \frac{1\hbox { kg}}{1,000 \hbox{ g}}=8.1$$ kg.
The cook has $$25$$ kg of potatoes, which is about $$3$$ times what is needed.