Find the unknown number. If half of a number is added to 2 5 \frac25 , the result is the same as subtracting 1 10 \frac{1}{10} from the number.

Solve this problem, we're going to use our five steps for solving word problems. Where our first step is to understand the problem by reading the problem, drawing a picture when needed, and identifying and naming our unknown. So let's read this problem. If half of a number is added to two fifths, the result is the same as subtracting one tenth from that number. Drawing a picture isn't really going to help us here. So let's go ahead and identify name our unknowns. We have half of a number, so that's our unknown, and we'll call it x. Subtracting one tenth from the number, so that's the same number. So we're going to also call that x. So next, let's build our equation. And in order to do this, we're going to use some translation techniques that we've learned previously. So half of a number. Well, if our number is x, half of that number is x divided by two. So half of a number is added to two fifths. So let's take that and add two fifths. The result is the same. That's another way for saying equal to. The result is the same means equal to as subtracting one tenth from that number. So we have our number x and we're going to subtract one tenth from it. So now that we have our equation, we can solve it. Here we're going to be solving for x. So my first step in trying to isolate x is I'm going to go ahead and add one tenth to both sides. And in order to combine these two fractions, we need to have common denominators. So I'm going to multiply two fifths by two. So that way they both have a denominator of 10. So I have x over two plus two fifths times two is four tenths plus one tenth equals x. And then we're going to combine our like terms here and move our x over two to the other side by subtracting it. Simplifying, we get four tenths plus one tenth is five tenths, and 1x minus one half x is one half x. So further solving for x, we're going to multiply both sides by two, and we get 10 over 10 equals x. And 10 over 10 simply reduces to one, so we have x equals one. Last, let's check our answer. We can do this by taking x equals one and plugging it back into the equation that we built at the beginning. So one over two plus two fifths equals one minus one tenth. In order to add these two fractions, we need to get common denominators again. So we're going to multiply this one by five and multiply this one by two, and we get five tenths plus four tenths. And this is equal to one minuteus one tenth. So five tenths plus four tenths is nine tenths, and one minus one tenth is also nine tenths. So we can feel confident that our answer here is correct.

3. Solving Word Problems

Introduction to Problem Solving

Beginning Algebra

3. Solving Word Problems

Introduction to Problem Solving

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Multiple Choice

Find the unknown number.
If half of a number is added to $\frac{2}{5}$ , the result is the same as subtracting $\frac{1}{10}$ from the number.

$- \frac{1}{5}$

$negative 1$

$1$

$- \frac{3}{5}$

Verified step by step guidance

Let the unknown number be represented by the variable $x$.

Translate the problem statement into an equation: "If half of a number is added to two fifths, the result is the same as subtracting one tenth from the number." This can be written as: $\frac{1}{2}x + \frac{2}{5} = x - \frac{1}{10}$.

To solve for $x$, first eliminate the fractions by finding the least common denominator (LCD) of 2, 5, and 10, which is 10. Multiply every term in the equation by 10 to clear the denominators: \$10 \times \left(\frac{1}{2}x + \frac{2}{5}\right) = 10 \times \left(x - \frac{1}{10}\right)$.

Simplify both sides after multiplying: \$10 \times \frac{1}{2}x = 5x$, $10 \times \frac{2}{5} = 4$, $10 \times x = 10x$, and $10 \times \frac{1}{10} = 1$. So the equation becomes $5x + 4 = 10x - 1$.

Next, isolate the variable $x$ by moving all $x$ terms to one side and constants to the other: subtract \$5x$ from both sides and add $1$ to both sides to get $4 + 1 = 10x - 5x$, which simplifies to $5 = 5x$. Then, divide both sides by 5 to solve for $x$.

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Find the unknown number.
If half of a number is added to $\frac{2}{5}$ , the result is the same as subtracting $\frac{1}{10}$ from the number.