Q65. - Algebra 2

Question

Mitena is making two types of cookies. The first recipe calls for $2\frac{1}{4}$cups of flour, and the second calls for $1\frac{1}{8}$ cups of flour. If Mitena wants to make 3 batches of the first recipe and 2 batches of the second recipe, how many cups of flour will she need? Use the properties of real numbers to show how Mitena could compute this amount mentally. Justify each step.

Step-by-Step Solution

Answer

Number of cups of flour required is 9. 

Step 1 - Conversion of mix fraction

Convert mix fraction into improper fraction as shown below

$\begin{array}{c}2\frac{1}{4}=2+\frac{1}{4}\\ =\frac{8+1}{4}\\ =\frac{9}{4}\end{array}$

$\begin{array}{c}1\frac{1}{8}=1+\frac{1}{8}\\ =\frac{8+1}{8}\\ =\frac{9}{8}\end{array}$

Step 2 - Calculate number of cups of flour required

Number of cups required to make 1 batch of first recipe is $\frac{9}{4}$ 

To find number of cups required to make 3 batch of first recipe, multiply 3 with $\frac{9}{4}$ as shown below

$3·\frac{9}{4}=\frac{27}{4}$

Number of cups required to make 1 batch of second recipe is $\frac{9}{8}$ 

To find number of cups required to make 2 batch of second recipe, multiply 2 with $\frac{9}{8}$ as shown below

$2·\frac{9}{8}=\frac{9}{4}$

To evaluate total number of cups of flour required add the terms

$\begin{array}{c}\frac{27}{4}+\frac{9}{4}=\frac{27+9}{4}\\ =\frac{36}{4}\\ =9\end{array}$

Step 3 - Properties of real number

Properties of real numbers are written in following table:

For any real numbers a, b, and c:
Property	Addition	Multiplication
Commutative	a+b=b+a	$a·b=b·a$
Associative	$\left(a+b\right)+c=a+\left(b+c\right)$	$\left(a·b\right)·c=a·\left(b·c\right)$
Identity	a+0=a=0+a	$a·1=a=1·a$
Inverse	$a+\left(-a\right)=0=\left(-a\right)+a$	If $a\ne 0$, then $a·\frac{1}{a}=1=\frac{1}{a}·a$
Distributive	$a\left(b+c\right)=ab+ac\text{ and }\left(b+c\right)a=ba+ca$

Step 4 - Solve using mental calculation

To evaluate number (N) of cups required mentally write mix fraction as sum of integer and proper fraction then use properties of real number as shown below

$\begin{array}{c}N=3\left(2\frac{1}{4}\right)+2\left(1\frac{1}{8}\right)\\ =3\left(2+\frac{1}{4}\right)+2\left(1+\frac{1}{8}\right)\left(\text{Definition of mix fraction}\right)\\ =\left(3\cdot 2\right)+\left(3\cdot \frac{1}{4}\right)+\left(2\cdot 1\right)+\left(2\cdot \frac{1}{8}\right)\left(\text{Distributive}\right)\\ =6+\frac{3}{4}+2+\frac{1}{4}\left(\text{Simplification of parentheses}\right)\\ =6+2+\frac{3}{4}+\frac{1}{4}\left(\text{Commutative property of Addition}\right)\\ =8+\frac{3+1}{4}\left(\text{addition}\right)\\ =8+\frac{4}{4}\\ =8+1\\ =9\left(\text{addition}\right)\end{array}$