Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Simplify \(\dfrac{2a}{5}\) + \(\dfrac{a}{3}\).
Show worked solution
\(\dfrac{2a}{5}\) = 6a/15 and \(\dfrac{a}{3}\) = 5a/15, so the sum is \(\dfrac{11a}{15}\). 3a/8 adds the tops and the bottoms; 3a/15 changes the bottoms without scaling the tops; 2a2/15 multiplies the fractions instead of adding.
2 Write \(\dfrac{x + 5}{2}\) + \(\dfrac{x + 1}{4}\) as one fraction with denominator 4.
Show worked solution
2(x + 5) + (x + 1) = 2x + 10 + x + 1 = 3x + 11, so the fraction is (3x + 11)/4.
3 Write \(\dfrac{n + 7}{3}\) + \(\dfrac{n - 1}{6}\) as one fraction with denominator 6. What is the number term in its numerator?
Show worked solution
2(n + 7) + (n − 1) = 2n + 14 + n − 1 = 3n + 13, so the fraction is (3n + 13)/6 and the number term is 13.
Level 2 · Core
4 Simplify \(\dfrac{3x - 1}{4}\) − \(\dfrac{x - 3}{6}\).
Show worked solution
3(3x − 1) − 2(x − 3) = 9x − 3 − 2x + 6 = 7x + 3, giving \(\dfrac{7x + 3}{12}\). \(\dfrac{7x - 9}{12}\) is the sign error −2 × −3 = −6; \(\dfrac{7x + 2}{12}\) only multiplies the first term of each numerator; \(\dfrac{2x + 2}{12}\) forgets to scale the numerators at all.
5 Write \(\dfrac{2m + 3}{5}\) − \(\dfrac{m - 4}{2}\) as one fraction with denominator 10.
Show worked solution
2(2m + 3) − 5(m − 4) = 4m + 6 − 5m + 20 = −m + 26, so the fraction is (−1m + 26)/10, that is (26 − m)/10.
6 Simplify \(\dfrac{15x - 10}{5}\) − \(\dfrac{8x - 12}{4}\), then find its value when x = 9.
Show worked solution
\(\dfrac{15x - 10}{5}\) = 3x − 2 and \(\dfrac{8x - 12}{4}\) = 2x − 3. So (3x − 2) − (2x − 3) = x + 1, and with x = 9 the value is 10. Check: 125/5 − 60/4 = 25 − 15 = 10 ✓.
Level 3 · Challenge
7 Three statements:
1 \(\dfrac{y - 1}{2}\) − \(\dfrac{y - 1}{7}\) = \(\dfrac{5(y - 1)}{14}\)
2 \(\dfrac{4x + 12}{4}\) = x + 12
3 \(\dfrac{x}{2}\) − \(\dfrac{x - 6}{3}\) = \(\dfrac{x + 12}{6}\)
Which statements are correct?
1 \(\dfrac{y - 1}{2}\) − \(\dfrac{y - 1}{7}\) = \(\dfrac{5(y - 1)}{14}\)
2 \(\dfrac{4x + 12}{4}\) = x + 12
3 \(\dfrac{x}{2}\) − \(\dfrac{x - 6}{3}\) = \(\dfrac{x + 12}{6}\)
Which statements are correct?
Show worked solution
7(y − 1) − 2(y − 1) = 5(y − 1), over 14 (1 ✓). 4x + 12 = 4(x + 3), so it simplifies to x + 3, not x + 12 (2 ✗). 3x − 2(x − 6) = 3x − 2x + 12 = x + 12, over 6 (3 ✓). 1 and 3.
8 \(\dfrac{2x + a}{4}\) − \(\dfrac{x - 2}{3}\) = \(\dfrac{bx + 23}{12}\) for every x. Find the numbers a and b.
Show worked solution
3(2x + a) − 4(x − 2) = 6x + 3a − 4x + 8 = 2x + 3a + 8. Matching x: b = 2. Matching numbers: 3a + 8 = 23, so 3a = 15 and a = 5.
9 \(\dfrac{x}{3}\) + \(\dfrac{x + 1}{4}\) − \(\dfrac{x - 2}{6}\) is written as one fraction \(\dfrac{ax + b}{12}\). What is a + b?
Show worked solution
4x + 3(x + 1) − 2(x − 2) = 4x + 3x + 3 − 2x + 4 = 5x + 7, so a = 5, b = 7 and a + b = 12.