Solve For A A A In The Equation:$\frac 2}{7} \times \frac{a}{7} = \frac{10}{49}$10. Stretch Your Thinking Ginny Went To The Store With A Gift Card. She Spent $\frac{2 {5}$ Of The Amount On Her Gift Card. Then, She Spent

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Introduction

In this article, we will delve into solving for aa in the equation 27×a7=1049\frac{2}{7} \times \frac{a}{7} = \frac{10}{49}. This equation involves fractions and multiplication, making it a great example of how to solve for a variable in a complex equation. We will break down the solution step by step, making it easy to follow and understand.

Understanding the Equation

The given equation is 27×a7=1049\frac{2}{7} \times \frac{a}{7} = \frac{10}{49}. To solve for aa, we need to isolate the variable aa on one side of the equation. The equation involves multiplication of fractions, so we will start by simplifying the left-hand side of the equation.

Simplifying the Left-Hand Side

To simplify the left-hand side of the equation, we can multiply the two fractions together. When multiplying fractions, we multiply the numerators together and the denominators together.

27×a7=2×a7×7\frac{2}{7} \times \frac{a}{7} = \frac{2 \times a}{7 \times 7}

This simplifies to:

2a49\frac{2a}{49}

So, the equation becomes:

2a49=1049\frac{2a}{49} = \frac{10}{49}

Isolating the Variable aa

Now that we have simplified the left-hand side of the equation, we can isolate the variable aa by getting rid of the fraction on the left-hand side. We can do this by multiplying both sides of the equation by the reciprocal of the fraction, which is 492\frac{49}{2}.

2a49×492=1049×492\frac{2a}{49} \times \frac{49}{2} = \frac{10}{49} \times \frac{49}{2}

This simplifies to:

a=102a = \frac{10}{2}

Solving for aa

Now that we have isolated the variable aa, we can solve for its value. We can simplify the fraction on the right-hand side by dividing the numerator by the denominator.

a=102=5a = \frac{10}{2} = 5

Therefore, the value of aa is 5.

Conclusion

In this article, we solved for aa in the equation 27×a7=1049\frac{2}{7} \times \frac{a}{7} = \frac{10}{49}. We simplified the left-hand side of the equation by multiplying the two fractions together, and then isolated the variable aa by getting rid of the fraction on the left-hand side. Finally, we solved for the value of aa by simplifying the fraction on the right-hand side. The value of aa is 5.

Additional Practice Problems

If you want to practice solving for variables in equations, here are a few additional problems:

  • 34×b4=1516\frac{3}{4} \times \frac{b}{4} = \frac{15}{16}
  • 23×c3=2027\frac{2}{3} \times \frac{c}{3} = \frac{20}{27}
  • 12×d2=58\frac{1}{2} \times \frac{d}{2} = \frac{5}{8}

Try solving these problems on your own, and then check your answers with the solutions below.

Solutions

  • 34×b4=1516\frac{3}{4} \times \frac{b}{4} = \frac{15}{16}
    • 3b16=1516\frac{3b}{16} = \frac{15}{16}
    • b=5b = 5
  • 23×c3=2027\frac{2}{3} \times \frac{c}{3} = \frac{20}{27}
    • 2c9=2027\frac{2c}{9} = \frac{20}{27}
    • c=10c = 10
  • 12×d2=58\frac{1}{2} \times \frac{d}{2} = \frac{5}{8}
    • d4=58\frac{d}{4} = \frac{5}{8}
    • d=10d = 10

Stretch Your Thinking: Ginny's Gift Card

Ginny went to the store with a gift card. She spent 25\frac{2}{5} of the amount on her gift card. Then, she spent 34\frac{3}{4} of the remaining amount. What fraction of the original amount did Ginny spend in total?

Step 1: Find the Fraction of the Original Amount Spent on the First Purchase

Ginny spent 25\frac{2}{5} of the original amount on her first purchase. To find the fraction of the original amount spent on the first purchase, we can multiply the fraction of the original amount spent by the fraction of the remaining amount.

25×45=825\frac{2}{5} \times \frac{4}{5} = \frac{8}{25}

Step 2: Find the Fraction of the Original Amount Spent on the Second Purchase

Ginny spent 34\frac{3}{4} of the remaining amount on her second purchase. To find the fraction of the original amount spent on the second purchase, we can multiply the fraction of the remaining amount spent by the fraction of the original amount.

34×35=920\frac{3}{4} \times \frac{3}{5} = \frac{9}{20}

Step 3: Find the Total Fraction of the Original Amount Spent

To find the total fraction of the original amount spent, we can add the fractions of the original amount spent on the first and second purchases.

825+920=64100+45100=109100\frac{8}{25} + \frac{9}{20} = \frac{64}{100} + \frac{45}{100} = \frac{109}{100}

Therefore, Ginny spent 109100\frac{109}{100} of the original amount in total.

Conclusion

In this article, we solved for aa in the equation 27×a7=1049\frac{2}{7} \times \frac{a}{7} = \frac{10}{49}. We simplified the left-hand side of the equation by multiplying the two fractions together, and then isolated the variable aa by getting rid of the fraction on the left-hand side. Finally, we solved for the value of aa by simplifying the fraction on the right-hand side. The value of aa is 5. We also solved a problem involving Ginny's gift card, where we found that she spent 109100\frac{109}{100} of the original amount in total.

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Introduction

In our previous article, we solved for aa in the equation 27×a7=1049\frac{2}{7} \times \frac{a}{7} = \frac{10}{49}. We also solved a problem involving Ginny's gift card, where we found that she spent 109100\frac{109}{100} of the original amount in total. In this article, we will answer some frequently asked questions (FAQs) related to solving for variables in equations and more.

Q&A

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Once you have the GCD, you can divide both the numerator and the denominator by the GCD to simplify the fraction.

Q: How do I multiply fractions?

A: To multiply fractions, you need to multiply the numerators together and the denominators together. For example, if you have 23×45\frac{2}{3} \times \frac{4}{5}, you would multiply the numerators together to get 2×4=82 \times 4 = 8, and multiply the denominators together to get 3×5=153 \times 5 = 15. The result would be 815\frac{8}{15}.

Q: How do I divide fractions?

A: To divide fractions, you need to invert the second fraction and multiply. For example, if you have 23÷45\frac{2}{3} \div \frac{4}{5}, you would invert the second fraction to get 54\frac{5}{4}, and then multiply to get 23×54=1012\frac{2}{3} \times \frac{5}{4} = \frac{10}{12}. You can then simplify the fraction to get 56\frac{5}{6}.

Q: How do I solve for a variable in an equation?

A: To solve for a variable in an equation, you need to isolate the variable on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same value.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to multiply the numerator and the denominator by the reciprocal of the denominator. For example, if you have 2345\frac{\frac{2}{3}}{\frac{4}{5}}, you would multiply the numerator and the denominator by the reciprocal of the denominator to get 23×54=1012\frac{2}{3} \times \frac{5}{4} = \frac{10}{12}. You can then simplify the fraction to get 56\frac{5}{6}.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to solving for variables in equations and more. We covered topics such as simplifying fractions, multiplying and dividing fractions, solving for variables in equations, and simplifying complex fractions. We hope that this article has been helpful in answering your questions and providing you with a better understanding of these topics.

Additional Practice Problems

If you want to practice solving for variables in equations, here are a few additional problems:

  • 34×b4=1516\frac{3}{4} \times \frac{b}{4} = \frac{15}{16}
  • 23×c3=2027\frac{2}{3} \times \frac{c}{3} = \frac{20}{27}
  • 12×d2=58\frac{1}{2} \times \frac{d}{2} = \frac{5}{8}

Try solving these problems on your own, and then check your answers with the solutions below.

Solutions

  • 34×b4=1516\frac{3}{4} \times \frac{b}{4} = \frac{15}{16}
    • 3b16=1516\frac{3b}{16} = \frac{15}{16}
    • b=5b = 5
  • 23×c3=2027\frac{2}{3} \times \frac{c}{3} = \frac{20}{27}
    • 2c9=2027\frac{2c}{9} = \frac{20}{27}
    • c=10c = 10
  • 12×d2=58\frac{1}{2} \times \frac{d}{2} = \frac{5}{8}
    • d4=58\frac{d}{4} = \frac{5}{8}
    • d=10d = 10

Stretch Your Thinking: Ginny's Gift Card (Again)

Ginny went to the store with a gift card. She spent 25\frac{2}{5} of the amount on her gift card. Then, she spent 34\frac{3}{4} of the remaining amount. What fraction of the original amount did Ginny spend in total?

Step 1: Find the Fraction of the Original Amount Spent on the First Purchase

Ginny spent 25\frac{2}{5} of the original amount on her first purchase. To find the fraction of the original amount spent on the first purchase, we can multiply the fraction of the original amount spent by the fraction of the remaining amount.

25×45=825\frac{2}{5} \times \frac{4}{5} = \frac{8}{25}

Step 2: Find the Fraction of the Original Amount Spent on the Second Purchase

Ginny spent 34\frac{3}{4} of the remaining amount on her second purchase. To find the fraction of the original amount spent on the second purchase, we can multiply the fraction of the remaining amount spent by the fraction of the original amount.

34×35=920\frac{3}{4} \times \frac{3}{5} = \frac{9}{20}

Step 3: Find the Total Fraction of the Original Amount Spent

To find the total fraction of the original amount spent, we can add the fractions of the original amount spent on the first and second purchases.

825+920=64100+45100=109100\frac{8}{25} + \frac{9}{20} = \frac{64}{100} + \frac{45}{100} = \frac{109}{100}

Therefore, Ginny spent 109100\frac{109}{100} of the original amount in total.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to solving for variables in equations and more. We covered topics such as simplifying fractions, multiplying and dividing fractions, solving for variables in equations, and simplifying complex fractions. We hope that this article has been helpful in answering your questions and providing you with a better understanding of these topics.