Estimating Quotients worksheet for practicing division estimation skills.

Here are the solutions for the estimating quotients problems.

Step-by-Step Solution:

To estimate a quotient, we look for two numbers close to the dividend (the first number) that are easy to divide by the divisor (the second number). Usually, these are multiples of the divisor found in multiplication tables.

2. $51 \div 6$
* Think of multiples of 6 close to 51.
* $6 \times 8 = 48$
* $6 \times 9 = 54$
* 51 is between 48 and 54.
* The quotient is between 8 and 9.
* Since 51 is closer to 48 (difference of 3) than to 54 (difference of 3), it is right in the middle. We can estimate 8 or 9. Let's pick 8 as a safe lower estimate, or often textbooks accept the closer whole number. 51 is slightly closer to 48? No, $51-48=3$ and $54-51=3$. It is exactly in the middle. Let's look at problem 1 for guidance. Problem 1 ($23 \div 4$) used 24 ($4\times6$) and 20 ($4\times5$). 23 is closer to 24, so they estimated 6. Here, 51 is equidistant. However, usually, we round to the nearest whole number. Let's look at the next ones to see if there's a clearer pattern. Let's stick with finding the closest multiple.
* Actually, let's re-evaluate "between". The question asks for two numbers the quotient lies between. Then estimate.
* Range: 8 and 9.
* Estimate: 8 (or 9). Let's provide 8 as it's a standard floor estimate, but 9 is also valid. Let's look at #3.

3. $76 \div 9$
* Multiples of 9: $9 \times 8 = 72$, $9 \times 9 = 81$.
* 76 is between 72 and 81.
* The quotient is between 8 and 9.
* 76 is closer to 72 (diff 4) than 81 (diff 5).
* Estimated quotient is 8.

4. $154 \div 12$
* Multiples of 12: $12 \times 10 = 120$, $12 \times 11 = 132$, $12 \times 12 = 144$, $12 \times 13 = 156$.
* 154 is very close to 156.
* The quotient is between 12 and 13.
* Since 154 is extremely close to 156, the estimated quotient is 13.

5. $109 \div 7$
* Multiples of 7: $7 \times 10 = 70$, $7 \times 15 = 105$, $7 \times 16 = 112$.
* 109 is between 105 and 112.
* The quotient is between 15 and 16.
* 109 is closer to 105 (diff 4) than 112 (diff 3)? No, $109-105=4$ and $112-109=3$. It is closer to 112.
* Estimated quotient is 16.

6. $125 \div 8$
* Multiples of 8: $8 \times 10 = 80$, $8 \times 15 = 120$, $8 \times 16 = 128$.
* 125 is between 120 and 128.
* The quotient is between 15 and 16.
* 125 is closer to 128 (diff 3) than 120 (diff 5).
* Estimated quotient is 16.

7. $225 \div 16$
* Multiples of 16: $16 \times 10 = 160$, $16 \times 14 = 224$ ($16 \times 10 = 160, 16 \times 4 = 64, 160+64=224$).
* $16 \times 15 = 240$.
* 225 is between 224 and 240.
* The quotient is between 14 and 15.
* 225 is very close to 224.
* Estimated quotient is 14.

8. $210 \div 13$
* Multiples of 13: $13 \times 10 = 130$, $13 \times 16 = 208$ ($13 \times 10=130, 13 \times 6=78, 130+78=208$).
* $13 \times 17 = 221$.
* 210 is between 208 and 221.
* The quotient is between 16 and 17.
* 210 is very close to 208 (diff 2).
* Estimated quotient is 16.

9. $105 \div 12$
* Multiples of 12: $12 \times 8 = 96$, $12 \times 9 = 108$.
* 105 is between 96 and 108.
* The quotient is between 8 and 9.
* 105 is closer to 108 (diff 3) than 96 (diff 9).
* Estimated quotient is 9.

10. $342 \div 15$
* Multiples of 15: $15 \times 20 = 300$, $15 \times 22 = 330$, $15 \times 23 = 345$.
* 342 is between 330 and 345.
* The quotient is between 22 and 23.
* 342 is closer to 345 (diff 3) than 330 (diff 12).
* Estimated quotient is 23.

Final Answer:
2. Between 8 and 9; Estimated quotient is 8
3. Between 8 and 9; Estimated quotient is 8
4. Between 12 and 13; Estimated quotient is 13
5. Between 15 and 16; Estimated quotient is 16
6. Between 15 and 16; Estimated quotient is 16
7. Between 14 and 15; Estimated quotient is 14
8. Between 16 and 17; Estimated quotient is 16
9. Between 8 and 9; Estimated quotient is 9
10. Between 22 and 23; Estimated quotient is 23