Exe 1C

BASIC LEVEL

1. Estimation and Calculation for 218÷31

Question: Seema evaluates 218÷31 using a calculator and she says that the answer is 70.3. Without doing the actual calculation, use estimation to determine whether Seema’s answer is reasonable. Then use a calculator to evaluate 218÷31. Is your estimated value close to the actual value?

Estimation

To estimate 218÷31, we round both numbers to make the division simple:

• Round 218 to the nearest hundred or a compatible number, 210 or 200.
• Round 31 to the nearest ten, 30.

Using 210÷30:

Estimated Value=30210​=7

Is Seema’s answer reasonable?

No, Seema’s answer of 70.3 is not reasonable. Our estimated value is 7. The actual answer should be close to 7, not 70. Seema likely made a calculation error, perhaps by multiplying instead of dividing, or entering 2180÷31.

Actual Calculation

Using a calculator:

Actual Value=31218​≈7.0322…

Comparison:

Our estimated value of 7 is very close to the actual value of 7.0322….

2. Estimation and Calculation of Expressions

Question: Estimate each of the following without using a calculator. Then use a calculator to evaluate each of the following. Are your estimated values close to the actual values?

(a) 2013×39

Estimation:

Round to the nearest compatible numbers: 2013≈2000 and 39≈40.

Estimated Value=2000×40=80,000

Actual Calculation:

Using a calculator:

Actual Value=2013×39=78,507

Comparison:

The estimated value of 80,000 is close to the actual value of 78,507.

(b) [Katex]145.6 ÷ \sqrt{65.4}[\Katex]​

Estimation:

145.6 ÷ \sqrt{65.4}

Round to the nearest compatible numbers:

• 145.6≈150
• 65.4 is very close to 64, and 64​=8.
  Estimated Value=150÷8
  Using simple division: 160÷8=20.
  So, 150÷8 is just under 20.
  Estimated Value≈18.75 (Or simply 19)

Actual Calculation:

Using a calculator:

Actual Value=145.6÷65.4​≈145.6÷8.0870…≈18.005

Comparison:

The estimated value of 18.75 (or 19) is close to the actual value of 18.005.

3. Significant Figures and Estimation

Question:

(i) Express 3.612 and 29.87 correct to 2 significant figures.

(ii) Use your answers in part (i) to estimate the value of 3.612÷29.87.

(i) Express to 2 Significant Figures (2 s.f.)

The first significant figure (s.f.) is the first non-zero digit. We look at the digit after the 2nd s.f. to round.

• 3.612:
  • 1st s.f. is 3. 2nd s.f. is 6.
  • The digit after 6 is 1. Since 1<5, we round down.3.6 (2 s.f.)
• 29.87:
  • 1st s.f. is 2. 2nd s.f. is 9.
  • The digit after 9 is 8. Since 8≥5, we round up the 9, which carries over to the 2.30 (2 s.f.)

(ii) Estimate 3.612÷29.87

Use the rounded values from part (i):

Estimated Value=3.6÷30

Estimated Value=0.12

4. Car Travel Estimation

Question: A car travels 274 km. It travels an average of 9.1 km on a litre of petrol. Write down a calculation you could do mentally to estimate the number of litres used.

Goal: Estimate Total Distance÷Distance per Litre.

Estimation:

Round the values to compatible numbers:

• 274 km is close to 270 km.
• 9.1 km/L is close to 9 km/L.

Calculation:

Estimated Litres=9270​

Estimated Litres=30

5. Square Tablecloth Length

Question: Mariam wants to buy a tablecloth that covers her square tabletop completely. If the area of the tablecloth is 6400 square centimetres, what is the length of the side of the table?

/k

Reasoning:

Since the tabletop is square, the side length (s) is the square root of the area (A).

s=A s = \sqrt{A}​

Calculation:

s=6400​s=64×100​s=64​×100​s=8×10s=80 cm
s=A s = \sqrt{6400}
s=A s = \sqrt{64 x 1000}​​

6. Smallest Number to Make a Perfect Square

Question: What is the smallest number that will divide into 216 to get a perfect square?

Step 1: Find the prime factorization of 216.

216=2×108216=2×2×54216=2×2×2×27216=2×2×2×3×3×3216=23×33

Step 2: Identify the factors with odd exponents.

For a number to be a perfect square, all exponents in its prime factorization must be even. In 23×33, both 2 and 3 have an odd exponent (3).

Step 3: Determine the smallest divisor.

To make the exponents even, we need to divide by the product of the base numbers that have odd exponents.

Divisor=2×3=6

Verification:

216÷6=36

Since 36=62, which is a perfect square, the smallest number is 6.

7. Gardener’s Plants in a Square Formation

Question: A gardener plants 25 trees in a square formation. If he plants 25 trees in a row, how many trees are there?

Interpretation: The question is poorly phrased. It says he plants trees in a square formation AND that he plants 25 trees in a row. This must mean the formation is a square with 25 trees on each side.

Calculation:

For a square formation with s trees in a row, the total number of trees is s2.

Total Trees=25×25=625

8. Ratio of Areas (Estimation)

Question: Estimate the ratio of the area of the shaded region to that of the unshaded region in the figure.

Reasoning:

The image shows a larger, roughly rectangular/square region with an irregularly shaped shaded region within it. We must visually estimate the fraction of the total area covered by the shaded part.

1. Visually divide the entire figure into smaller, equal blocks.
2. The shaded region appears to occupy approximately 1 small block.
3. The unshaded region appears to occupy approximately 3 small blocks.
4. The ratio of Shaded Area:Unshaded Area is approximately 1:3.

Estimated Ratio=1:3

9. Arranging Numbers in Ascending Order

Question: Arrange the following numbers in ascending order.

(a) −1.8,7.25,−1.5,0.5,13.1,−14.12

(b) 134.7,−13.47,1.34,0.134,−0.134,1347

Ascending order means from smallest (most negative) to largest (most positive).

(a) −1.8,7.25,−1.5,0.5,13.1,−14.12

1. Identify the smallest negative: −14.12
2. Compare remaining negatives: −1.8 and −1.5.
3. Arrange the positives: 0.5,7.25,13.1.

−14.12,−1.8,−1.5,0.5,7.25,13.1

(b) 134.7,−13.47,1.34,0.134,−0.134,1347

1. Identify the smallest negative: −13.47
2. Compare remaining negatives: −0.134.
3. Arrange the positives: 0.134,1.34,134.7,1347.

−13.47,−0.134,0.134,1.34,134.7,1347

10. Arranging Numbers in Descending Order

Question: Arrange the following numbers in descending order.

(a) −7.8,−18.2,−15.5,0,17.6,1.76

(b) 25.4,9.3,−9.3,0.13,−0.55,2.54

Descending order means from largest (most positive) to smallest (most negative).

(a) −7.8,−18.2,−15.5,0,17.6,1.76

1. Identify the largest positive: 17.6
2. Arrange the remaining positives/zero: 1.76,0.
3. Compare the negatives: −7.8,−15.5,−18.2.

17.6,1.76,0,−7.8,−15.5,−18.2

(b) 25.4,9.3,−9.3,0.13,−0.55,2.54

1. Identify the largest positive: 25.4
2. Arrange the remaining positives: 9.3,2.54,0.13.
3. Compare the negatives: −0.55,−9.3.

25.4,9.3,2.54,0.13,−0.55,−9.3

11. Finding Square Roots

Question:

(a) Find the square of the following numbers.
(i) 23
(ii) 212

(b) Find the square root of the following numbers.
(i) 361
(ii) 576
(iii) 16.81
(iv) 784
(v) 5.29
(vi) 2.89
(vii) 484
(viii) 65.61
(ix) 900/441

(a) Find the square of the following numbers.

(i) 23² = 529

(ii) 212² = 44,944

(b) Find the square root of the following numbers.

(i) 361​=19

(ii) 576​=24

(iii) 16.81​=4.1 (Since 412=1681)

(iv) 784​=28

(v) 5.29​=2.3 (Since 232=529)

(vi) 2.89​=1.7 (Since 172=289)

(vii) 484​=22

(viii) 65.61​=8.1 (Since 812=6561)

(ix) 441900​​=441​900​​=2130​=710​≈1.4286

12. Shopkeeper’s Order (Total Cost Estimation)

Question: A shopkeeper makes the following orders from a wholesaler:

Show how you estimate the total amount of money that the shopkeeper has to pay, giving your answer correct to the nearest hundred rupees.

Step 1: Estimate the cost for each item by rounding Quantity and Cost per item.

Step 2: Estimate the Total Amount.

Estimated Total=600+200+1500+1200+500=4000

Step 3: Round the estimate to the nearest hundred rupees.

The estimated total of 4000 is already a multiple of 100.

Estimated Total (to the nearest hundred rupees)=4000 PKR

13. Currency Conversion Estimation

Question: A bag costs Malaysian Ringgit (RM) 25. The conversion rate is RM1=PKR 52.2588. Without using a calculator, estimate the price of the bag in PKR.

Step 1: Round the conversion rate to 1 significant figure.

RM 1=PKR 52.2588≈PKR 50

Step 2: Estimate the total price.

Estimated Price in PKR=Cost in RM×Rounded RateEstimated Price=25×50Estimated Price=1250 PKR

14. Best Value Calculation

Question: Without using a calculator, decide which of the following options is better value for money.

Option A: 300 g Jelly Beans PKR 580

Option B: 500 g Jelly Beans PKR 990

Reasoning: Better value means the lowest cost per unit mass (PKR per 100 g). We estimate the cost per 100 g.

Option A: 300 g for PKR 580

Round the price: 580≈600.

Estimated Cost per 100 g=3600​=PKR 200

Option B: 500 g for PKR 990

Round the price: 990≈1000.

Estimated Cost per 100 g=51000​=PKR 200

Conclusion:

Since both options estimate to PKR 200 per 100 g, they are very similar in value. We check the exact values (mentally, 580/3≈193 and 990/5=198). Since PKR 580 is slightly cheaper for 300 g than PKR 600 would be, Option A is slightly better value.

Option A offers slightly better value for money (Estimated cost per 100 g is ≈PKR 193 vs PKR 198 for Option B).

15. Sale Discount Calculation

Question: Shop A sells a bottle of juice for PKR 79.50 with a 20% discount while Shop B sells the same bottle of juice for PKR 69.50 with a 10% discount. Write down a calculation you could do mentally to help you decide which shop to buy the juice from.

Goal: Estimate the final price for each shop.

Shop A: PKR 79.50 with 20% Discount

Estimation: Round PKR 79.50≈PKR 80.

A 20% discount means you pay 80% of the price.

Discount≈20% of 80=0.2×80=16

Estimated Final Price (A)≈80−16=PKR 64

Shop B: PKR 69.50 with 10% Discount

Estimation: Round PKR 69.50≈PKR 70.

A 10% discount means you pay 90% of the price.

Discount≈10% of 70=0.1×70=7

Estimated Final Price (B)≈70−7=PKR 63

Conclusion:

Since PKR 63 is less than PKR 64, Shop B is cheaper.

16. Handbag Currency Conversion

Question: A handbag costs 26 700 Korean Won (KRW). The conversion rate is KRW 1=PKR 0.18. Without using a calculator, estimate the price of the handbag in PKR.

Step 1: Round the amount and the rate to 1 significant figure.

• 26 700 KRW≈30 000 KRW
• PKR 0.18≈PKR 0.2

Step 2: Estimate the price in PKR.

Estimated Price=Rounded KRW×Rounded RateEstimated Price=30 000×0.2Estimated Price=30 000×102​

Estimated Price=3 000×2=6000

The estimated price of the handbag in PKR is 6000 PKR