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How to Solve Proportions with Ease: A Step-by-Step Guide with Lesson 7 Homework Practice Answer Key



Lesson 7 Homework Practice Solving Proportions Answer Key and Tips




Proportions are statements that two ratios are equal. For example, if you have a recipe that calls for 2 cups of flour and 3 eggs, and you want to make half of the recipe, you can write a proportion to find out how much flour and eggs you need:




Lesson 7 Homework Practice Solving Proportions Answer Key



$$\frac23=\fracx1.5$$


In this proportion, x is the amount of flour you need for half of the recipe, and 1.5 is half of 3 eggs. To solve a proportion, you can use cross-multiplication, which means multiplying the cross products and setting them equal. For example:


$$2\times 1.5=3\times x$$


$$3=3x$$


$$x=1$$


So you need 1 cup of flour for half of the recipe.


In this article, we will show you how to solve proportions using cross-multiplication, and provide you with the answer key and tips for Lesson 7 Homework Practice Solving Proportions.


How to Solve Proportions Using Cross-Multiplication




To solve proportions using cross-multiplication, follow these steps:


  • Identify the two ratios that are equal in the proportion.



  • Multiply the numerator of one ratio by the denominator of the other ratio, and write the product on one side of an equation.



  • Multiply the denominator of one ratio by the numerator of the other ratio, and write the product on the other side of the equation.



  • Solve for the unknown variable by isolating it on one side of the equation.



  • Check your answer by plugging it back into the original proportion and simplifying.



Here is an example of how to solve a proportion using cross-multiplication:


Solve for x: $$\frac45=\fracx15$$


  • The two ratios that are equal in the proportion are $$\frac45$$ and $$\fracx15$$.



  • Multiply the numerator of one ratio by the denominator of the other ratio, and write the product on one side of an equation: $$4\times 15=60$$.



  • Multiply the denominator of one ratio by the numerator of the other ratio, and write the product on the other side of the equation: $$5\times x=5x$$.



  • Solve for x by isolating it on one side of the equation: $$60=5x \Rightarrow \frac605=x \Rightarrow x=12$$.



  • Check your answer by plugging it back into the original proportion and simplifying: $$\frac45=\frac1215 \Rightarrow \frac4\times 35\times 3=\frac1215 \Rightarrow \frac1215=\frac1215 \Rightarrow \textTrue$$.



Lesson 7 Homework Practice Solving Proportions Answer Key and Tips




Here are the answers and tips for Lesson 7 Homework Practice Solving Proportions. You can use cross-multiplication to solve each problem, or any other method that works for you.


ProblemAnswerTip


Solve for x: $$\frac68=\fracx16$$x = 12You can simplify $$\frac68$$ to $$\frac34$$ before cross-multiplying.


Solve for x: $$\fracx9=\frac1027$$x = 3You can simplify $$\frac1027$$ to $$\frac10 \div 1027 \div 10=\frac12.7$$ before cross-multiplying.


Solve for x: $$\fracx+25=\fracx-14$$x = 18You can distribute 5 and 4 to both sides of the equation after cross-multiplying: $$(x+2)\times 4=5\times (x-1) \Rightarrow 4x+8=5x-5$$. Then solve for x by subtracting 4x from both sides and adding 5 to both sides: $$8=x-5 \Rightarrow x=13$$.


Solve for x: $$\fracx-36=\fracx+912$$x = 15You can distribute 6 and 12 to both sides of the equation after cross-multiplying: $$(x-3)\times


12=6\times (x+9) \Rightarrow


12x-36=6x+54$$. Then solve


for x by subtracting


6x from both sides


and adding


36 to both sides:


$$48=x


\Rightarrow


x=48$$.


Solve for x: $$\fracx+49=\fracx-26$$<


x = -6You can distribute


9 and


6 to both sides


of


the equation after


cross-multiplying:


$$(x+4)\times


6=9\times (x-2)


\Rightarrow


6x+24=9x-18$$. Then solve


for x by subtracting


9x from both sides


and subtracting


24 from both sides:


$$-42=-3x


\Rightarrow


x=14$$.


Solve for x: $$\fracx-78=\fracx+110$$<


x = -39You can distribute


8 and


10 to both sides


of


the equation after


cross-multiplying:


$$(x-7)\times


10=8\times (x+1)


\Rightarrow


10x-70=8x+8$$. Then solve


for x by subtracting


10x from both sides


and adding


70 to both sides:


$$78=-2x


\Rightarrow


x=-39$$.


Solve for x: $$\fracx+37=\fracx-511$$<


x = -16.5You can distribute


7 and


11 to both sides


of


the equation after


cross-multiplying:


$$(x+3)\times


11=7\times (x-5)


\Rightarrow


11x+33=7x-35$$. Then solve


for x by subtracting


11x from both sides


and subtracting


33 from both sides:


$$-68=-4x


\Rightarrow


x=17$$.


Solve for x: $$\fracx-23=\fracx+49$$<


x = -6You can distribute


3 and


9 to both sides


of


the equation after


cross-multiplying:


$$(x-2)\times


9=3\times (x+4)


\Rightarrow


9x-18=3x+12$$. Then solve


for x by subtracting


9x from both sides


and adding


18 to both sides:


$$30=-6x


\Rightarrow


x=-5$$.


Solve for x: $$\fracx+812=\fracx-416$$<


x = -32


How to Apply Proportions to Real-World Problems




Proportions can be used to solve many real-world problems involving rates, ratios, and percentages. For example, you can use proportions to find out how much of an ingredient you need for a recipe, how much time it takes to travel a certain distance, how much interest you earn on an investment, or how much tax you pay on a purchase.


To apply proportions to real-world problems, follow these steps:


  • Read the problem carefully and identify the given information and the unknown quantity.



  • Write a ratio that compares two quantities with the same units in the problem.



  • Write another ratio that compares two quantities with the same units in the problem, one of which is the unknown quantity.



  • Write a proportion by setting the two ratios equal to each other.



  • Solve the proportion using cross-multiplication or any other method that works for you.



  • Check your answer by plugging it back into the original proportion and simplifying.



  • Write your answer in a complete sentence with appropriate units.



Here are some examples of how to apply proportions to real-world problems:


Example 1: A recipe for chocolate chip cookies calls for 2 cups of flour and 1 cup of chocolate chips. How many cups of chocolate chips do you need if you use 3 cups of flour?


  • The given information is 2 cups of flour and 1 cup of chocolate chips. The unknown quantity is how many cups of chocolate chips you need if you use 3 cups of flour.



  • A ratio that compares two quantities with the same units in the problem is $$\frac\textflour\textchocolate chips=\frac21$$.



  • Another ratio that compares two quantities with the same units in the problem, one of which is the unknown quantity, is $$\frac\textflour\textchocolate chips=\frac3x$$, where x is the unknown quantity.



  • A proportion by setting the two ratios equal to each other is $$\frac21=\frac3x$$.



  • To solve the proportion using cross-multiplication, we multiply the numerator of one ratio by the denominator of the other ratio, and write the product on one side of an equation. Then we multiply the denominator of one ratio by the numerator of the other ratio, and write the product on the other side of the equation. Then we solve for x by isolating it on one side of the equation: $$2\times x=1\times 3 \Rightarrow 2x=3 \Rightarrow \frac2x2=\frac32 \Rightarrow x=\frac32$$.



  • To check our answer by plugging it back into the original proportion and simplifying, we substitute x with $$\frac32$$ and see if both sides are equal: $$\frac21=\frac3\frac32 \Rightarrow \frac21=\frac3\times 23 \Rightarrow \frac21=\frac63 \Rightarrow \frac21=\frac6 \div 33 \div 3 \Rightarrow \frac21=\frac21 \Rightarrow \textTrue$$.



  • To write our answer in a complete sentence with appropriate units, we say: You need $$\frac32$$ cups of chocolate chips if you use 3 cups of flour.



Example 2: A car travels 120 miles in 2 hours. How long will it take to travel 180 miles at the same speed?


  • The given information is 120 miles in 2 hours and 180 miles. The unknown quantity is how long it will take to travel 180 miles at the same speed.



  • A ratio that compares two quantities with the same units in the problem is $$\frac\textmiles\texthours=\frac1202$$.



  • Another ratio that compares two quantities with the same units in the problem, one of which is the unknown quantity, is $$\frac\textmiles\texthours=\frac180x$$, where x is the unknown quantity.



  • A proportion by setting the two ratios equal to each other is $$\frac1202=\frac180x$$.



To solve the proportion using cross-multiplication, we multiply


the numerator of one ratio by


the denominator of


the other ratio,


and write


the product on


one side


of an equation.


Then we multiply


the denominator


of one ratio by


the numerator


of


the other ratio,


and write


the product on


the other side


of


the equation.


Then we solve


for x by isolating


it on one side


of


the equation:


$$120\times x=2\times 180 \Rightarrow


120x=360 \Rightarrow


\frac120x120=\frac360120


\Rightarrow


  • x=3$$.



To check our answer by plugging it back into


the original proportion and simplifying,


we substitute x with


3 and see if both sides are equal:


$$\frac1202=\frac1803


\Rightarrow


\frac120 \div 602 \div 60=\frac180 \div 603 \div 60


\Rightarrow


\frac20.033=


\frac30.05


\Rightarrow


\frac2\times 0.050.033\times 0.05=


\frac3\times 0.0330.05\times 0.033


\Rightarrow


\frac0.10.00165=


\frac0.0990.00165


\Rightarrow


60=60


\Rightarrow


  • \textTrue$$.



  • To write our answer in a complete sentence with appropriate units, we say: It will take 3 hours to travel 180 miles at the same speed.



Conclusion




Proportions are useful tools to solve many real-world problems involving rates, ratios, and percentages. In this article, we have shown you how to solve proportions using cross-multiplication, and provided you with the answer key and tips for Lesson 7 Homework Practice Solving Proportions. We hope this article has helped you understand and apply proportions better. Remember to check your answers by plugging them back into the original proportions and simplifying. Happy solving! a27c54c0b2


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