Question: 11 The table shows the costs of sending a parcel by the Express service and by the Rapid service.
Type of service | Cost
Express | £15.21
Rapid | £17.28
Julia has to send 12 parcels.
It will be cheaper to send the parcels by the Express service than by the Rapid service.
(a) How much cheaper?
Luke wants to send 21 parcels by the Express service.
He does the calculation 20 × £15 = £300 to estimate the cost
(b) Explain why Luke's calculation shows the actual cost will be more than £300
1. **State the problem:**
Julia needs to send 12 parcels. We want to find how much cheaper it is to send them by Express service compared to Rapid service.
2. **Given data:**
Cost per parcel by Express service = $15.21$
Cost per parcel by Rapid service = $17.28$
Number of parcels Julia sends = $12$
3. **Calculate total cost for each service:**
- Express total cost = $12 \times 15.21$
- Rapid total cost = $12 \times 17.28$
4. **Calculate the difference (how much cheaper):**
$$\text{Difference} = 12 \times 17.28 - 12 \times 15.21 = 12 \times (17.28 - 15.21)$$
5. **Simplify inside the parentheses:**
$$17.28 - 15.21 = 2.07$$
6. **Calculate the difference:**
$$12 \times 2.07 = 24.84$$
So, sending 12 parcels by Express service is $24.84$ cheaper than by Rapid service.
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7. **For part (b):**
Luke estimates the cost for 21 parcels by calculating $20 \times 15 = 300$.
8. **Explain why this is an underestimate:**
- The actual cost per parcel is $15.21$, not $15$.
- Luke used $20$ parcels instead of $21$.
9. **Calculate the actual cost:**
$$21 \times 15.21 = 21 \times (15 + 0.21) = 21 \times 15 + 21 \times 0.21 = 315 + 4.41 = 319.41$$
10. **Conclusion:**
Luke's calculation of $300$ is less than the actual cost $319.41$ because he used a lower price per parcel and fewer parcels in his estimate.