Chapter 5.5, Problem 16HP

To determine

To find:Thegallons of water for $6$ cups of flour and the value of constant of proportionality.

Answer to Problem 16HP

The water needed for $6$ cups of flour is $1.5\text{gallons}$ andconstant of proportionalityis $4$ .

Explanation of Solution

Given information: The paper mache paste recipe needs flour for one cup of water is $\frac{1}{4}\text{c}$

Calculation:

Two quantities, in which the ratio or rate is constant, are to be said proportional and if the rates are not constant, then the quantities are said to be non-proportional. The constant ratio which is defining whether the two quantities are said to be proportional or not is known as constant of proportionality.

The given unit conversion of cups of Flour to cup of water in fraction is:

  $\frac{\frac{1}{4}\text{cups}\text{of}\text{flour}}{1\text{cup}\text{of}\text{water}}$

The unit conversion for $6$ cups of flour to $x$ cups of water in fraction is:

  $\frac{6\text{cups}\text{of}\text{flour}}{x\text{cup}\text{of}\text{water}}$

Now, write the above expression in proportion.

  $\frac{\frac{1}{4}\text{cups}\text{of}\text{flour}}{1\text{cup}\text{of}\text{water}}=\frac{6\text{cups}\text{of}\text{flour}}{x\text{cup}\text{of}\text{water}}$

To obtain the value of $x$ , apply cross multiplication in the above expression.

  $\begin{array}{c}\frac{x\text{cup}\text{of}\text{water}}{1\text{cup}\text{of}\text{water}}=\frac{6\text{cups}\text{of}\text{flour}}{\frac{1}{4}\text{cups}\text{of}\text{flour}}\\ x\text{cups}\text{of}\text{water}=\text{6}\times 4\text{cups}\text{of}\text{flour}\\ \text{=24}\text{cups}\text{of}\text{flour}\end{array}$

The formula for conversion of cups into gallons is:

  $1\text{c}=0.0625\text{gal}$

To find the conversion of $24\text{c}$ , multiply both sides of above conversion by $24$ .

  $\begin{array}{c}24\times 1\text{c}=24\times 0.0625\text{gal}\\ \text{=1}\text{.5}\text{gal}\end{array}$

The expression of Proportional relationships having constant of proportionality $k$

in equation form is:

  $y=kx$

To obtain the constant rate, substitute $6\text{cups}\text{of}\text{flour}$ for $x$ and $24\text{cups}\text{of}\text{water}$ for $y$ in the expression of proportional relationship.

  $\begin{array}{c}24\text{cups}\text{of}\text{water}=k\times 6\text{cups}\text{of}\text{flour}\\ k=\frac{24\text{cups}\text{of}\text{water}}{6\text{cups}\text{of}\text{flour}}\\ =4\end{array}$

Therefore, the water needed for $6$ cups of flour is $1.5\text{gallons}$ and constant of proportionality is $4$ .