Simplify:[(2⁄3) + (4⁄5 ÷ 2⁄7)] ÷ [(3⁄2)−(5⁄6 × 3⁄4)]

Correct — A. Finish each bracket, then divide the two results.

Left bracket:
4⁄5 ÷ 2⁄7 = 4⁄5 × 7⁄2 = 14⁄5
2⁄3 + 14⁄5 = 10⁄15 + 42⁄15 = 52⁄15

Right bracket:
5⁄6 × 3⁄4 = 15⁄24 = 5⁄8
3⁄2 − 5⁄8 = 12⁄8 − 5⁄8 = 7⁄8

Divide: 52⁄15 ÷ 7⁄8 = 52⁄15 × 8⁄7 = 416⁄105 → option (a)

Why the others are wrong

• (b)
  418⁄110 — 418⁄110 reduces to 19⁄5 = 3.8. The true value, 416⁄105, is about 3.96, so this near miss fails on exact working.
• (c)
  425⁄250 — 425⁄250 = 1.7. The left bracket alone is 52⁄15 ≈ 3.47, and dividing by 7⁄8, a number below 1, makes it larger, not smaller.
• (d)
  526⁄350 — 526⁄350 ≈ 1.50, below the left bracket's 3.47. Dividing 3.47 by 7⁄8, which is less than 1, must give more than 3.47.

Order of operations with fractions: finish each bracket first, and inside a bracket do ÷ and × before + and −. Here the inner parentheses already group the division and the product.

Dividing by a fraction means multiplying by its reciprocal: a⁄b ÷ c⁄d = a⁄b × d⁄c. Adding or subtracting needs a common denominator: 15 for thirds and fifths, 8 for halves and eighths.

Key facts

• a⁄b ÷ c⁄d = a⁄b × d⁄c.
• Dividing by a positive number below 1 gives a result larger than the number divided.
• Reduce an option before comparing: 418⁄110 = 19⁄5 and 425⁄250 = 17⁄10.

Common traps

• Multiplying instead of dividing: 4⁄5 × 2⁄7 = 8⁄35, which drags the whole left bracket down.
• Taking 418⁄110 for the answer because it looks close, when it reduces to 19⁄5 = 3.8.

No directly related past PYQ was found.