Frank's Ice Cream Hut had Chocolate or Vanilla shakes. On the 4 th of July they sold 1,386 shakes total with the ratio of chocolate to vanilla boing $$ 5: 9 $$. How many chocolar shakes did they sell?

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Let's denote the number of chocolate shakes sold as $$ x $$ and the number of vanilla shakes sold as $$ y $$.

We are given that the total number of shakes sold is 1,386, so we have the equation:
$$ x + y = 1386 $$

We are also given that the ratio of chocolate to vanilla shakes is $$ 5:9 $$, which means:
$$ \frac{x}{y} = \frac{5}{9} $$

We can solve these two equations simultaneously to find the number of chocolate shakes sold.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x+y=1386\\frac{x}{y}=\frac{5}{9}\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=1386-y\\frac{x}{y}=\frac{5}{9}\end{array}\right.$$
- step2: Substitute the value of $$x:$$
  $$\frac{1386-y}{y}=\frac{5}{9}$$
- step3: Cross multiply:
  $$\left(1386-y\right)\times 9=y\times 5$$
- step4: Simplify the equation:
  $$9\left(1386-y\right)=5y$$
- step5: Expand the expression:
  $$12474-9y=5y$$
- step6: Move the variable to the left side:
  $$12474-9y-5y=0$$
- step7: Subtract the terms:
  $$12474-14y=0$$
- step8: Move the constant to the right side:
  $$-14y=0-12474$$
- step9: Remove 0:
  $$-14y=-12474$$
- step10: Change the signs:
  $$14y=12474$$
- step11: Divide both sides:
  $$\frac{14y}{14}=\frac{12474}{14}$$
- step12: Divide the numbers:
  $$y=891$$
- step13: Substitute the value of $$y:$$
  $$x=1386-891$$
- step14: Simplify:
  $$x=495$$
- step15: Calculate:
  $$\left\{ \begin{array}{l}x=495\y=891\end{array}\right.$$
- step16: Check the solution:
  $$\left\{ \begin{array}{l}x=495\y=891\end{array}\right.$$
- step17: Rewrite:
  $$\left(x,y\right) = \left(495,891\right)$$
The solution to the system of equations is $$ x = 495 $$ and $$ y = 891 $$.

Therefore, Frank's Ice Cream Hut sold 495 chocolate shakes on the 4th of July.
Frank's Ice Cream Hut sold 495 chocolate shakes on the 4th of July.

Frank’s Ice Cream Hut had Chocolate or Vanilla shakes. On the 4 th of July they sold
1,386 shakes total with the ratio of chocolate to vanilla boing . How many chocolar
shakes did they sell?

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Frank’s Ice Cream Hut had Chocolate or Vanilla shakes. On the 4 th of July they sold 1,386 shakes total with the ratio of chocolate to vanilla boing . How many chocolar shakes did they sell?