NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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A craft designer in Brisbane is creating a pattern using two types of beads in the ratio 78:51. If the designer uses a total of 442 beads, how many of the first type of bead are used?

Express your answer as an exact fraction.

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Two red counters for every three blue counters. Red:blue = 2:3. There are 5 counters in all, so the red fraction of the whole is 2/5.RedRedBlueBlueBlue2 red : 3 blue · 5 counters in all
Red:blue = 2:3. There are 5 counters in all, so the red fraction of the whole is 2/5.
Translate a situation into a model. With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.$4 + $2/kmDefine tariff→C = 4 + 2dBuild rule→d=3 → C=$10Interpret
With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.

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Worked solution

  1. The ratio of the two types of beads is 78:51. This means for every 78 beads of the first type, there are 51 beads of the second type.
  2. The total number of parts in the ratio is 78 + 51 = 129 parts.
  3. The total number of beads used is 442. This total corresponds to 129 parts.
  4. To find the number of beads per part, divide the total beads by the total parts: 442 ÷ 129.
  5. Calculate 442 ÷ 129. This gives 442/129 beads per part.
  6. The number of the first type of bead is 78 parts. Multiply 78 by the beads per part: 78 × (442/129).
  7. Perform the multiplication: (78 × 442) / 129.
  8. Calculate 78 × 442: 78 × 400 = 31,200; 78 × 42 = 3,276; total = 31,200 + 3,276 = 34,476.
  9. Now divide 34,476 by 129: 34,476 ÷ 129.
  10. Simplify 34,476 / 129. First, divide numerator and denominator by 3: 34,476 ÷ 3 = 11,492; 129 ÷ 3 = 43. So, 11,492 / 43.
  11. Now divide 11,492 by 43: 43 × 267 = 43 × 200 = 8,600; 43 × 67 = 2,881; total = 8,600 + 2,881 = 11,481. Difference: 11,492 - 11,481 = 11. So 267 + 11/43 = 267 11/43. But since we are looking for an exact fraction, keep as 11,492 / 43.
  12. The number of the first type of bead is 11,492 / 43. This fraction is already in simplest form because 43 is prime and does not divide 11,492 evenly. Therefore, the answer is 11,492/43 beads.

Reference answer: 11492/43

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