Writing Algebraic Expressions Worksheets - Math Monks - Free Printable

Let’s go step by step through each problem. We’ll write algebraic expressions for phrases, and then write phrases for given expressions.

---

Part 1: Write an expression for each phrase

We’ll use a variable (like *x* or *y*) to represent “a number” unless told otherwise.

---

1) Six less than a number

“Less than” means subtraction, but note: “six less than a number” means you start with the number and subtract 6.

→ So if the number is *x*, it’s x - 6

---

2) The sum of 10 times a number and two

“Sum” = addition
“10 times a number” = 10*x*
“And two” = + 2

→ So: 10x + 2

---

3) The quotient of 20 divided by y, increased by seven

“Quotient” = division → 20 ÷ y = 20/y
“Increased by seven” = add 7

→ So: 20/y + 7

*(Note: Since they used “y”, we keep using y here.)*

---

4) The product of 4 and a number, divided by 2

“Product” = multiplication → 4 × x = 4x
“Divided by 2” → divide the whole thing by 2

→ So: (4x)/2 or simplified: 2x

But since the question says “write an expression”, either is fine — but let’s keep it as written: (4x)/2

Actually, in most cases, simplifying is okay unless told not to. But let’s check the wording: “the product... divided by 2” — so yes, (4x)/2 is correct. We can also write it as 4x/2.

But to be precise and match the phrasing, we’ll write: 4x / 2

(Though mathematically, that equals 2x — but again, unless asked to simplify, we leave it as is.)

Wait — actually, looking at other problems, they don’t require simplification unless obvious. Let’s stick to direct translation.

So: 4x ÷ 2 or \frac{4x}{2}

In standard algebra notation, we write fractions with numerator over denominator.

But since this is text-based, we can write: (4x)/2

However, in many curricula, they accept 4x/2 without parentheses because multiplication/division are left-to-right, but to be safe, let’s use parentheses.

Actually, no — order of operations: multiplication and division have same precedence, so 4x/2 is interpreted as (4x)/2 anyway.

So we can safely write: 4x/2

But let me double-check: “product of 4 and a number” → 4x; then “divided by 2” → divide that result by 2 → yes, 4x/2.

Alternatively, some might write \frac{4x}{2}, but since we’re writing linearly, 4x/2 is fine.

I think for consistency with how others are written, we’ll use slash notation.

---

5) The difference of ten and six, multiplied by a number

“Difference of ten and six” = 10 - 6 = 4
Then “multiplied by a number” → 4 × x = 4x

BUT — wait! Is it (10 - 6) × x? Yes.

The phrase says: “the difference of ten and six, [then] multiplied by a number”

So first compute difference, then multiply.

→ So: (10 - 6)x or just 4x

Again, should we simplify? The problem doesn’t say to simplify, but “difference of ten and six” is clearly 4, so multiplying by x gives 4x.

But to show the structure, maybe (10 - 6)x?

Looking at similar problems, often they expect the unsimplified form if it shows the operation.

But here, “difference of ten and six” is a constant — it’s always 4. So probably they want 4x.

Wait — let’s read carefully: “The difference of ten and six, multiplied by a number”

This could be ambiguous, but grammatically, “the difference...” is the subject being multiplied.

So: (10 - 6) × x = 4x

I think 4x is acceptable and simplest.

But to be thorough, let’s see what the expected answer is likely to be.

In many textbooks, they would write 4x for this.

Alternatively, if they wanted to emphasize the steps, they might write (10-6)x, but I think 4x is fine.

Actually, let’s compare to #2: “sum of 10 times a number and two” → 10x + 2, which is simplified.

Similarly, here, difference of 10 and 6 is 4, so 4x.

I’ll go with 4x

But wait — another interpretation: could it mean “ten minus (six multiplied by a number)”? No, because it says “the difference of ten and six” — that’s a single entity — then “multiplied by a number”.

So yes, (10-6)*x = 4x.

Final decision: 4x

---

Now Part 2: Write a phrase for each expression.

---

6) 2x - 6

This is “two times a number, minus six” or “six less than twice a number”

Both are valid.

Common phrasing: “Six less than twice a number”

Because 2x is “twice a number”, then minus 6 → “six less than...”

Yes.

So: Six less than twice a number

---

7) 5(6 + x)

This is 5 times the quantity (6 plus x)

So: “Five times the sum of six and a number”

Or: “The product of five and the sum of six and a number”

Simplest: Five times the sum of six and a number

---

8) (2 + x) - 8

First, 2 + x is “the sum of two and a number”

Then subtract 8: “that sum minus eight” or “eight less than the sum of two and a number”

Better: Eight less than the sum of two and a number

Because (2+x) - 8 = same as -8 + (2+x), but “less than” reverses order.

“A less than B” means B - A.

So “eight less than (sum)” means (sum) - 8 → perfect.

So: Eight less than the sum of two and a number

---

9) (2 + x)/5

This is “the sum of two and a number, divided by five”

Or: “the quotient of the sum of two and a number and five”

Simpler: The sum of two and a number, divided by five

---

10) 20/x + 7

First, 20/x is “twenty divided by a number” or “the quotient of twenty and a number”

Then +7: “increased by seven” or “plus seven”

So: The quotient of twenty and a number, increased by seven

Note: This matches exactly with problem #3, which was “The quotient of 20 divided by y, increased by seven” → 20/y + 7

Here it’s 20/x + 7, so same thing, just different variable.

So phrase: The quotient of twenty and a number, increased by seven

---

Now, let’s list all answers clearly.

Final Answers:

1) x - 6
2) 10x + 2
3) 20/y + 7
4) 4x / 2 (or 2x, but I'll keep as 4x/2 to match phrasing)
Wait — reconsidering #4.

Problem 4: “The product of 4 and a number, divided by 2”

If we write 4x/2, it’s correct, but perhaps they expect it simplified? In many worksheets, they do simplify.

Looking back at problem 5, we simplified 10-6 to 4.

Similarly, 4x/2 simplifies to 2x.

And 2x is simpler and still accurate.

Moreover, in problem 6, they have 2x - 6, so 2x is used.

I think for consistency, we should simplify where possible.

So for #4: 2x

Similarly, for #5: 4x

Let me confirm with logic:

#4: product of 4 and x is 4x, divided by 2 is 2x → yes.

#5: difference of 10 and 6 is 4, times x is 4x → yes.

So I’ll adjust:

4) 2x
5) 4x

That makes more sense.

Also, for #3, they used y, so we keep y.

For others, we use x as default.

Now, finalizing:

──────────────────────────────────────

Final Answer:
1) x - 6
2) 10x + 2
3) 20/y + 7
4) 2x
5) 4x
6) Six less than twice a number
7) Five times the sum of six and a number
8) Eight less than the sum of two and a number
9) The sum of two and a number, divided by five
10) The quotient of twenty and a number, increased by seven