What Times 1.5 Is 70

What Times 1.5 is 70? Unlocking the Power of Algebra

Finding the unknown in a simple equation might seem like a basic math problem, but understanding how to solve "What times 1.5 is 70?" unlocks a fundamental concept in algebra: solving for an unknown variable. This seemingly simple question isn't just about getting the right answer; it's about grasping the underlying principles that empower you to tackle more complex mathematical challenges. This article will guide you through several methods to solve this problem, explaining the logic behind each approach and expanding on the broader mathematical concepts involved.

Understanding the Problem: Translating Words into Equations

The question "What times 1.5 is 70?" can be easily translated into an algebraic equation. Let's represent the unknown number with the variable 'x'. The question then becomes:

1.5 * x = 70

This equation states that 1.5 multiplied by an unknown number (x) equals 70. Our goal is to isolate 'x' and find its value.

Method 1: Using Division – The Direct Approach

The most straightforward way to solve for x is to use division. Since 1.5 is multiplied by x, we can undo this operation by dividing both sides of the equation by 1.5:

1.5 * x / 1.5 = 70 / 1.5

This simplifies to:

x = 70 / 1.5

Now, we perform the division:

x ≈ 46.67

Therefore, the number that, when multiplied by 1.5, equals 70 is approximately 46.67. The "approximately" is used because the division results in a repeating decimal.

Method 2: Converting to Fractions – For Enhanced Understanding

Working with fractions can sometimes offer a clearer understanding of the underlying mathematical principles. Let's convert 1.5 to a fraction:

1.5 = 3/2

Our equation now becomes:

(3/2) * x = 70

To isolate x, we multiply both sides by the reciprocal of 3/2, which is 2/3:

(2/3) * (3/2) * x = 70 * (2/3)

This simplifies to:

x = 140/3

Converting this improper fraction to a mixed number or decimal gives us:

x = 46 2/3 or x ≈ 46.67

This method confirms our previous result, highlighting the equivalence between decimal and fractional representations.

Method 3: Using Proportions – A Visual Approach

Proportions offer a visual and intuitive way to solve this type of problem. We can set up a proportion as follows:

1.5 / 1 = 70 / x

This proportion reads: "1.5 is to 1 as 70 is to x". To solve for x, we cross-multiply:

1.5 * x = 70 * 1

1.5x = 70

Now, we divide both sides by 1.5, which leads us back to the same solution:

x ≈ 46.67

This method provides a different perspective on the problem, illustrating the relationship between the known and unknown values.

Method 4: Trial and Error – A Basic Approach (Less Efficient)

While not the most efficient method for this specific problem, trial and error can be a useful approach for beginners or for simple estimations. You could start by multiplying 1.5 by different numbers until you get close to 70. While this method might eventually lead to the correct answer, it's less systematic and time-efficient compared to the other methods discussed.

Expanding on the Concepts: Beyond the Specific Problem

Solving "What times 1.5 is 70?" is more than just finding a numerical answer. It illustrates several crucial mathematical concepts:

• Algebraic Equations: The problem showcases the fundamental principle of solving algebraic equations, where the goal is to isolate the unknown variable.
• Inverse Operations: To solve for x, we utilized inverse operations. Multiplication and division are inverse operations; they undo each other.
• Equivalence: The principle of equivalence states that if you perform the same operation on both sides of an equation, the equality remains true. This is the basis for all the solution methods described above.
• Fractions and Decimals: The problem highlights the interchangeability of fractions and decimals, reinforcing the understanding of different number representations.
• Proportions: The use of proportions provides a visual and intuitive way to grasp the relationship between different quantities.

Frequently Asked Questions (FAQ)

• Q: Is the answer exactly 46.67?

  • A: No, the answer is approximately 46.67. The exact answer is 46 2/3, which is a repeating decimal (46.6666...).

• Q: Can I use a calculator to solve this?

  • A: Absolutely! Calculators are useful tools for performing calculations, especially for more complex problems. However, understanding the underlying mathematical principles is crucial, even when using technology.

• Q: What if the problem used a different number instead of 70?

  • A: The same methods would apply. You would simply substitute the new number into the equation and follow the steps to solve for x.

• Q: Are there other ways to solve this type of problem?

  • A: Yes, more advanced techniques exist, particularly for more complex equations, including graphical methods and iterative numerical methods. However, the methods discussed here are sufficient for this particular problem.

Conclusion: Mastering the Fundamentals

Solving "What times 1.5 is 70?" might seem trivial at first glance. However, by understanding the different approaches to solving this seemingly simple equation, you solidify your understanding of fundamental algebraic concepts. This foundational knowledge is essential for tackling more advanced mathematical challenges in the future. Remember, the key isn't just getting the right answer; it's understanding why the answer is correct and grasping the broader mathematical principles involved. This comprehensive approach ensures a deeper and more enduring understanding of mathematics. Practice consistently, explore different methods, and you will confidently navigate the world of algebraic equations.