Worksheet featuring 20 multiplication problems with algebraic fractions, sourced from a Victorian textbook.

Let’s solve each problem one by one. We’re multiplying algebraic fractions by whole numbers. The key idea is: multiply the numerator by the whole number, then simplify if possible.

We’ll go row by row, left to right.

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1. $\frac{x + 1}{3} \times 6$

Multiply numerator: $(x + 1) \times 6 = 6x + 6$
Then divide by 3: $\frac{6x + 6}{3} = 2x + 2$

✔ Final: $2x + 2$

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2. $\frac{x + 2}{2} \times 4$

Numerator: $(x + 2) \times 4 = 4x + 8$
Divide by 2: $\frac{4x + 8}{2} = 2x + 4$

✔ Final: $2x + 4$

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3. $\frac{x + 4}{2} \times 6$

Numerator: $(x + 4) \times 6 = 6x + 24$
Divide by 2: $\frac{6x + 24}{2} = 3x + 12$

✔ Final: $3x + 12$

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4. $\frac{y + 5}{3} \times 12$

Numerator: $(y + 5) \times 12 = 12y + 60$
Divide by 3: $\frac{12y + 60}{3} = 4y + 20$

✔ Final: $4y + 20$

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5. $\frac{7 + x}{4} \times 8$

Numerator: $(7 + x) \times 8 = 56 + 8x$
Divide by 4: $\frac{56 + 8x}{4} = 14 + 2x$ → or $2x + 14$

✔ Final: $2x + 14$

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6. $\frac{2x + 1}{4} \times 8$

Numerator: $(2x + 1) \times 8 = 16x + 8$
Divide by 4: $\frac{16x + 8}{4} = 4x + 2$

✔ Final: $4x + 2$

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7. $\frac{2x + 3}{3} \times 6$

Numerator: $(2x + 3) \times 6 = 12x + 18$
Divide by 3: $\frac{12x + 18}{3} = 4x + 6$

✔ Final: $4x + 6$

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8. $\frac{5 + 6x}{4} \times 8$

Numerator: $(5 + 6x) \times 8 = 40 + 48x$
Divide by 4: $\frac{40 + 48x}{4} = 10 + 12x$ → or $12x + 10$

✔ Final: $12x + 10$

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9. $\frac{7 + 3x}{5} \times 15$

Numerator: $(7 + 3x) \times 15 = 105 + 45x$
Divide by 5: $\frac{105 + 45x}{5} = 21 + 9x$ → or $9x + 21$

✔ Final: $9x + 21$

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10. $\frac{2 + 7x}{1\frac{1}{2}} \times 3$

First, convert mixed number: $1\frac{1}{2} = \frac{3}{2}$

So we have: $\frac{2 + 7x}{\frac{3}{2}} \times 3$

Dividing by a fraction = multiply by reciprocal:

$\left( (2 + 7x) \times \frac{2}{3} \right) \times 3 = (2 + 7x) \times 2 = 4 + 14x$

✔ Final: $14x + 4$

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11. $\frac{x - 2}{5} \times 10$

Numerator: $(x - 2) \times 10 = 10x - 20$
Divide by 5: $\frac{10x - 20}{5} = 2x - 4$

✔ Final: $2x - 4$

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12. $\frac{y - 5}{3} \times 9$

Numerator: $(y - 5) \times 9 = 9y - 45$
Divide by 3: $\frac{9y - 45}{3} = 3y - 15$

✔ Final: $3y - 15$

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13. $\frac{7 - p}{2} \times 4$

Numerator: $(7 - p) \times 4 = 28 - 4p$
Divide by 2: $\frac{28 - 4p}{2} = 14 - 2p$ → or $-2p + 14$

✔ Final: $-2p + 14$ (or $14 - 2p$, both OK)

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14. $\frac{17 - 2p}{1\frac{1}{2}} \times 3$

Again, $1\frac{1}{2} = \frac{3}{2}$

So: $\frac{17 - 2p}{\frac{3}{2}} \times 3 = (17 - 2p) \times \frac{2}{3} \times 3 = (17 - 2p) \times 2 = 34 - 4p$

✔ Final: $-4p + 34$ (or $34 - 4p$)

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15. $\frac{2x - 1}{2} \times 8$

Numerator: $(2x - 1) \times 8 = 16x - 8$
Divide by 2: $\frac{16x - 8}{2} = 8x - 4$

✔ Final: $8x - 4$

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16. $\frac{3x - 2}{4} \times 8$

Numerator: $(3x - 2) \times 8 = 24x - 16$
Divide by 4: $\frac{24x - 16}{4} = 6x - 4$

✔ Final: $6x - 4$

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17. $\frac{5 + 6x}{7} \times 14$

Numerator: $(5 + 6x) \times 14 = 70 + 84x$
Divide by 7: $\frac{70 + 84x}{7} = 10 + 12x$ → or $12x + 10$

✔ Final: $12x + 10$

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18. $\frac{7 - 3x}{5} \times 15$

Numerator: $(7 - 3x) \times 15 = 105 - 45x$
Divide by 5: $\frac{105 - 45x}{5} = 21 - 9x$ → or $-9x + 21$

✔ Final: $-9x + 21$

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19. $\frac{8 - 2y}{34} \times 12$

Numerator: $(8 - 2y) \times 12 = 96 - 24y$
Divide by 34: $\frac{96 - 24y}{34}$

Simplify fraction: factor out 2 from numerator and denominator:

= $\frac{2(48 - 12y)}{2 \cdot 17} = \frac{48 - 12y}{17}$

Can’t simplify further. So leave as $\frac{48 - 12y}{17}$ or write as $-\frac{12y - 48}{17}$ — but usually we keep positive leading term in numerator if possible.

Actually, let’s check: 48 and 12 are divisible by... 12? But 17 is prime. So no common factors with 17.

Wait — maybe better to write as:

$\frac{-12y + 48}{17}$ or factor out -12? Not necessary.

But let’s see: original expression: $\frac{8 - 2y}{34} \times 12 = \frac{(8 - 2y) \cdot 12}{34}$

Factor numerator: $8 - 2y = 2(4 - y)$, so:

= $\frac{2(4 - y) \cdot 12}{34} = \frac{24(4 - y)}{34} = \frac{12(4 - y)}{17}$

That’s simpler!

So: $\frac{12(4 - y)}{17} = \frac{48 - 12y}{17}$ — same thing.

We can leave it as $\frac{48 - 12y}{17}$ or factor out 12: $\frac{12(4 - y)}{17}$ — either is fine. Since the question doesn’t specify, we’ll go with expanded form unless instructed otherwise.

But actually, looking at other answers, they all simplified to polynomials without denominators. Here, 34 doesn’t cancel fully with 12? Wait — 12 and 34 share factor 2.

Original: $\frac{8 - 2y}{34} \times 12 = \frac{(8 - 2y) \cdot 12}{34}$

Compute GCD of 12 and 34 is 2.

So: $\frac{(8 - 2y) \cdot 6}{17} = \frac{48 - 12y}{17}$

Yes, that’s simplest.

✔ Final: $\frac{48 - 12y}{17}$

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20. $\frac{7 - 7x}{9} \times 18$

Numerator: $(7 - 7x) \times 18 = 126 - 126x$
Divide by 9: $\frac{126 - 126x}{9} = 14 - 14x$ → or $-14x + 14$

Alternatively, factor first: $7 - 7x = 7(1 - x)$, so:

$\frac{7(1 - x)}{9} \times 18 = 7(1 - x) \times 2 = 14(1 - x) = 14 - 14x$

✔ Final: $-14x + 14$

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Now, compiling all final answers in order:

Final Answer:
1. $2x + 2$
2. $2x + 4$
3. $3x + 12$
4. $4y + 20$
5. $2x + 14$
6. $4x + 2$
7. $4x + 6$
8. $12x + 10$
9. $9x + 21$
10. $14x + 4$
11. $2x - 4$
12. $3y - 15$
13. $14 - 2p$
14. $34 - 4p$
15. $8x - 4$
16. $6x - 4$
17. $12x + 10$
18. $21 - 9x$
19. $\frac{48 - 12y}{17}$
20. $14 - 14x$