The document contains solutions to 4 math word problems: 1) Finding the value of a + b + c using the Pythagorean theorem. The answer is 47. 2) Balancing a scale by adding minimum number of balls. The answer is 5 balls. 3) Finding the common remainder when two numbers are divided by a 3-digit number. 4) Maximizing the product of a 5-digit number by placing digits 1-9 in the boxes.

1. In the figure below, PQRS is a rectangle. What is the value of a + b + c.
Solution:
Using Pythagoras Theorum, we know that:
𝑎2 = 𝑏2 + 92 --- ①
𝑐2
= 𝑏2
+ 162
--- ②
(9 + 16)2= 𝑎2 + 𝑐2 --- ③
And ① + ② = ③
2𝑏2
+ 92
+ 162
= 252
𝑏2
= ____
𝑏 = ____
Replace 𝑏 to ① + ②, we get 𝑎 and 𝑐.
Answer: 𝑎 + 𝑏 + 𝑐 = 47

2. If each large ball weighs 1
1
3
times the weight of each little ball, what is the
minimum number of balls that need to be added to the right-hand side to
make the scale balance? You may not remove balls, but only add small
and/or large balls to the right-hand side.
Solution:
𝑠𝑚𝑎𝑙𝑙 ∶ 𝑙𝑎𝑟𝑔𝑒 = 1 ∶ 1
1
3
= 3 ∶ 4
Which means we can assume the weight for
small ball is 3 and large ball is 4.
The weight for right-hand side = 2 𝑥 4 = 8
The weight for left-hand side = 9 𝑥 3 = 27
And we need to add __①__ to right-hand side.
Assuming we are adding 𝑎 small ball and 𝑏
large ball: 3𝑎 + 4𝑏 = __①__ --- ②
To get minimum number of balls, add as many large ball as possible. The
maximum large ball to add is 4.
From ②, 𝑎 = 1
Answer: 5

3. Solution:
Since the remainders are equal, this means the difference of 31513 and
344369, 344369 – 31513 = 2856, is divisible by the three-digit number.
You can use the division-by-prime method to obtain the factors for 2856.
2856 = 23 × 3 × 7 × 17
So the three-digit number can be 102, 119, 136, 168, 204, 238, 357, 408,
476, 714 and 952.
Use either one of these to find the remainder.
Answer: _____
When 31513 and 34369 are each divided by a certain three-digit number,
the remainders are equal. Find this remainder.

4. Solution:
Let’s rewrite the expression to 𝑎𝑏𝑐𝑑𝑒 × 𝑓𝑔ℎ × 𝑘
We know that the biggest digits 7, 8, 9 must go to 𝑎, 𝑓 and 𝑘;
5, 6 go to 𝑏 and 𝑔;
and 3, 4 go to 𝑐 and ℎ;
and 𝑑𝑒 = 21.
And we will get the largest product if the sum of the digits for 𝑎𝑏𝑐 and 𝑓𝑔ℎ are
the same.
Therefore, 𝑘 = 9, 𝑎𝑏𝑐 = 764 and 𝑓𝑔ℎ = 853
Answer: 76421 × 853 × 9
Fill the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 into the boxes
× ×
so that the expression will produce the largest product. (Each digit can be
used only once)