Solve: Two Times Sum Of Number And 5 Equals 20

Two times the sum ofa number and 5 is 20 is a straightforward algebraic statement that appears in many introductory math lessons. At its core, the phrase translates to the equation 2 × (x + 5) = 20, where x represents the unknown number. Solving this equation teaches the basic principles of distributing, isolating variables, and checking work—skills that form the foundation for more complex algebra later on. The following sections break down the problem step by step, explain why each move works, offer a real‑world analogy, provide practice opportunities, and highlight common pitfalls to avoid.

Understanding the Equation

Before jumping into calculations, it helps to rewrite the sentence in mathematical symbols. “Two times” signals multiplication by 2. “The sum of a number and 5” means we first add the unknown number (let’s call it x) to 5, giving x + 5. The word “is” tells us the result equals 20. Putting it together:

[ 2 \times (x + 5) = 20 ]

Parentheses are essential here because they indicate that the addition happens before the multiplication. If we ignored them and wrote 2 × x + 5 = 20, we would be solving a different problem altogether. The parentheses preserve the intended order of operations, ensuring we first find the sum, then double it.

Step‑by‑Step Solution

Solving the equation involves isolating x on one side. We can do this in two main ways: by dividing first or by distributing the 2 first. Both paths lead to the same answer, and showing both reinforces the flexibility of algebraic manipulation.

Method 1: Divide First

1. Divide both sides by 2 to undo the multiplication.
   [ \frac{2 \times (x + 5)}{2} = \frac{20}{2} ] Simplifying gives
   [ x + 5 = 10 ]

2. Subtract 5 from both sides to isolate x.
   [ x + 5 - 5 = 10 - 5 ] Hence
   [ x = 5 ]

Method 2: Distribute First

1. Apply the distributive property: multiply 2 by each term inside the parentheses.
   [ 2 \times x + 2 \times 5 = 20 ] Which becomes
   [ 2x + 10 = 20 ]

2. Subtract 10 from both sides.
   [ 2x + 10 - 10 = 20 - 10 ] Simplifying yields
   [ 2x = 10 ]

3. Divide by 2 to solve for x.
   [ \frac{2x}{2} = \frac{10}{2} ] So
   [ x = 5 ]

Both methods confirm that the unknown number is 5. To verify, substitute 5 back into the original statement: the sum of 5 and 5 is 10; two times 10 is 20, which matches the given result.

Why Each Step Works

Understanding the reasoning behind each operation prevents rote memorization and builds genuine comprehension.

• Division by 2 reverses the multiplication that was applied to the entire sum. Think of it as sharing a total equally into two parts; if two parts together make 20, each part must be 10.
• Subtraction of 5 undoes the addition that was performed inside the parentheses. If a number plus 5 equals 10, the original number must be 5 less than 10.
• Distribution spreads the multiplication over each addend, which is useful when you prefer to work with separate terms rather than a grouped sum. It relies on the distributive property a(b + c) = ab + ac, a fundamental rule of arithmetic.
• Final division isolates the variable by removing its coefficient. When 2x = 10, dividing both sides by 2 asks, “What number multiplied by 2 gives 10?” The answer is 5.

Real‑World Analogy

Imagine you are preparing gift bags for a party. Each bag must contain the same number of candies, and you decide to put 5 extra candies in every bag as a surprise. You have enough candy to fill exactly 20 bags, and you know that the total number of candies used is twice the amount you would have if you only counted the surprise candies plus the regular ones per bag. How many regular candies go in each bag?

Let x be the number of regular candies per bag. The surprise adds 5, so each bag holds x + 5 candies. Because you have 20 bags and the total candy used is twice the per‑bag amount, the relationship is 2 × (x + 5) = 20. Solving shows x = 5, meaning each bag gets 5 regular candies plus the 5 surprise candies, for a total of 10 candies per bag. This analogy mirrors the algebraic steps: first you halve the total to find the per‑bag amount, then you remove the surprise to find the regular portion.

Practice Problems

To solidify the concept, try solving similar equations on your own. Check your answers by substituting the solution back into the original statement.

1. Three times the sum of a number and 4 equals 33.
   [ 3(x + 4) = 33 ]

2. Half of the sum of a number and 7 is 9.
   [ \frac{1}{2}(x +

7. = 9 ]

Conclusion

Solving algebraic equations is a fundamental skill that extends far beyond the classroom. The methods we've explored – division, subtraction, distribution, and inverse operations – are not just tools for finding unknown variables; they are building blocks for understanding relationships between quantities and modeling real-world situations. By focusing on why each step works, we move beyond simple memorization and cultivate a deeper understanding of how mathematics can be used to solve problems and make sense of the world around us. Mastering these techniques empowers us to tackle increasingly complex mathematical challenges and appreciate the power of algebraic thinking in a wide range of disciplines. The ability to translate word problems into mathematical equations and then solve them provides a valuable framework for critical thinking and problem-solving skills applicable to numerous aspects of life.