MATHEMATICS ANSWERS, JULY 2020: 2020年7月的数学答案

Question # 1:  

Tom took 4 sweets, John took 3 sweets.  Mum paid $1.40 for the sweets and this price has a discount of 30% deducted.  How much is the marked-up price of 1 sweet? 

问题 # 1: 汤姆拿了4个糖果，约翰拿了3个糖果。妈妈花了$1.40买了这7个糖果，这个价格已经扣除了30％的折扣。  问题 #1: 1个糖果的加价是多少？

Answer # 1 - with all steps clearly stated.     答案 #1 - 明确说明所有步骤。

Let ‘ s ‘ is the price of one sweet after the 30% discount.  So, 7 ‘ s ‘ = $1.40;  ‘ s ‘ = $0.20

But there is a 30% discount.  So, ‘ s ‘ is = (100% - 30%) = 70% of the marked-up price ‘ SS ‘.

70% of ‘ SS  ‘ = $0.20. 100% of ‘ SS ‘ = $ (0.20 / 70 * 100) = $ 0.29 to the nearest cent.

答案 # 1 - 明确说明所有步骤。                    

设 “s” 为30％折扣后的一个糖果的价格。 因此，7 “s” = $ 1.40; “s”= $ 0.20.   

但是有30％的折扣。 因此，“s” = 标记价格 “SS” 的70％。

“SS” 的 70％ = $ 0.20.   “SS” 的 100％ = $（0.20 / 70 * 100）= $ 0.29 至最接近的一分。

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Question # 2:  

3 fountain pens cost $5.00 after 20% discount. 6 pencils cost $2.00 after 10% discount.

Find the cost of 10 fountains pens and 10 pencils, after discounts.

3支钢笔在20％折扣后的价格为 $5.00。 10％折扣后，六支铅笔的价格为$ 2.00。 

问题 #2: 打折后找到10支钢笔和10支铅笔的价格是多少？

Answer # 2 - with all steps clearly stated.    

3 fountain pens cost $5.00. So, 1 fountain pen costs $5.00 divided by 3 = $1.66.

But, this is after 20% discount. So, 80% of the fountain pen’s cost = $1.66

Each fountain pen (100%) costs $1.66 divided by 80% multiplied by 100%, which is equal to $2.08 (to the nearest cent).

Similarly,  6 pencils cost $2.00. So, 1 pencil costs $2.00 divided by 6 = $0.33

But, this is after a 10% discount. So, 90% of a pencil cost $0.33

Each pencil (100%) costs $0.33 divided by 90% multiplied by 100% equals $0.37 (to the nearest cent).

Finally, 10 fountain pens = 10 multiplied by $2.08 = $20.80. 10 pencils = 10 multiplied by $0.37 = $3.70. Total cost = $20.80 + $3.70 = $24.50

答案 # 2 - 明确说明所有步骤。                    

钢笔的价格为$ 5.00. 因此，一支钢笔的价格为 $5.00 除以 3 = $1.66。

但是，这是20％的折扣之后。 因此, 钢笔的80％价格为= $ 1.66

每支钢笔（100％）的价格为 $1.66除以80％乘以100％, 即等于$2.08（精确到最近的一分钱）。

同样, 6支铅笔的价格为 $2.00. 因此, 一支铅笔的价格为 $2.00除以6 = $0.33

但是, 这是在10％的折扣之后。因此, 一支 90％的铅笔售价为 $0.33

每支铅笔（100％）的售价为 $0.33除以 90％ 乘以100％ 等于$0.37（精确到最近的一分）。

最后，10支钢笔 = 10乘以$ 2.08 = $ 20.80. 10支铅笔 = 10乘以$ 0.37 = $ 3.70. 总售价 = $ 20.80 + $ 3.70 = $ 24.50

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Question # 3:

5 green apples and 14 red apples cost $40.00 after a 10% discount. The green apple costs twice as much as the red apples.  What is the total cost of 4 green apples plus 8 red apples?  What is the ‘marked-up’ price of the green apples and the red apples?

问题 # 3：

10％折扣后，5个绿色苹果和14个红色苹果的价格为$ 40.00。青苹果的价格是红苹果的两倍。 4个绿色苹果加8个红色苹果的总价格是多少？青苹果和红苹果的“表格“是多少？

Answer # 3 - with all steps clearly stated.     答案 # 3 - 明确说明所有步骤。

Let ‘Gp’ be the price of each Green apple after discount and ‘Rp’ be the price of each Red apple after discount.

The equations then are:

1. Gp = 2 x Rp
2. 5 Gp + 14 Rp = $40.00 or (5×2 Rp) + 14 Rp = $40.00
3. 24 Rp = 40.00; so Rp = 40 divided by 24 = $1.66
4. Each Red apple after discount is $1.66
5. Each Green apple after discount is twice as much = $3.32
6. 4 Gp and 8 Rp = (4 x $3.32) + (8 x $1.66) = $13.28 + $13.28 = $26.56

If GP = marked up price of Green Apple, and RP = marked up price of Red Apple, then:

7. There is a 10% discount for all apples.  So, 90% of GP = $3.32; 100% of GP = $3.32 divided by 90% multiplied by 100% = $3.69
8. Similarly, 90% of RP = $1.66; 100% of RP = $1.66 divided by 90% multiplied by 100% = $1.84 

假设“ Gp”是折扣后每个青苹果的价格，“ Rp”是折扣后每个红苹果的价格。 然后，等式为：

Gp = 2 x Rp

5 Gp + 14 Rp = $40.00 或（5×2 Rp）+ 14 Rp = $40.00

24 Rp = 40.00；因此Rp = 40除以24 = $ 1.66

折后每个红苹果是 $1.66

折扣后的每个青苹果价格翻倍 = $ 3.32

4 Gp 和 8 Rp =（4 x $ 3.32）+（8 x $ 1.66）= $ 13.28 + $ 13.28 = $ 26.56

如果GP = 青苹果的表价，而RP = 红苹果的表价，则：

所有苹果都有 10％的折扣。因此，GP的90％= $ 3.32； GP的100％= $ 3.32 除以 90％ 乘以100％= $ 3.69. 同样，RP的90％= $1.66; 100％的表价= $1.66除以90％乘以100％= $1.84.

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Question # 4:

Estimate the length of the bridge (line PR) in the diagrams shown below, if the river is 14 meters wide, and the two retaining walls are 28 metres and 35 meters high :

问题 # 3：如果河流为14米宽，而两个挡土墙分别为28米和35米高，则在下图中估算桥梁的长度（PR行）：

Answer # 4:

Line QS = Line PT = 14 meters.

In the triangle PRT, line RT = the distance of line RS minus the distance of line PQ. = (35 - 28) meters or 7 meters.

The bridge (line PR) is the hypotenuse of the right-angled triangle PRT. So the length of the bridge is the square root of √(7 squared plus 14 squared)

7 squared = 49; 14 squared = 196; Sum of the two = 245.

Answer is the square root of √245. We can do this by the division method as follows:

Answer = 15.66 meters.

Look at how this is done by viewing this YouTube video below: https://youtu.be/PJHtqMjrStk

答案 # 4 - 明确说明所有步骤。

线QS =线PT = 14米。

在三角形PRT中，线RT =线RS的距离减去线PQ的距离。 =（35 - 28）米或7米。
桥（线PR）是直角三角形PRT的斜边。所以桥的长度是（7 双倍 加14 双倍）的平方根。

7 双倍 = 49; 14 双倍 = 196; 两者之和= 245。
答案是245的平方根。我们可以通过除法执行以左操作：答案是 15.66 米。
观看观看下面的YouTube视频，了解操作方法：https：//youtu.be/PJHtqMjrStk 

Question # 5: Solve for ‘ y ‘ or what is the value of the unknown ‘ y ‘?

问题 # 5：求解 ‘y’ 或未知 ‘y’ 的值是多少？

3 ( 5y - 2y - 6) = 4 ( 2y + 5y -2)

Answer # 5: 答案 # 5。

Expanding, 打开括号. 15y - 6y - 18 = 8y + 20y - 8

Putting all the ‘ y ‘ on the left-hand side of the equal sign and all numbers on the right-hand side, we have:
将所有“ y”放在等号的左侧，将所有数字放在右侧，我们有：

15y - 6y - 20y = - 8 + 18

- (11 y) = 10 or 11y = -10

y = -10 / 11 or - 0.909

Question # 6: Solve for ‘ w ‘ and for ‘ y ‘. What are the values of ‘ w ‘ and of ‘ y ‘?

问题＃6：求解“ w”和“ y”。 “ w”和“ y”的值是多少？

3 (2w - 12) = (- 2) (2w - 7) y = 2(w + 5)

Answer # 6 : 答案 # 6。

Expand the equation # 1: 打开括号 - 方程式 # 1. 6w - 36 = - 4w + 14

‘ w ‘ to the left and numbers to the right. 左侧为“ w”，右侧为数字

6w + 4w = 14 + 36

10 w = 50

So w. = 5.

Expand the equation # 2: 打开括号 - 方程式 # 2. y = 2(w + 5)

y = 2w + 10

y = 2 x 5 + 10. =. 10 + 10. = 20.

Question # 7: Find the square root of (a) √144; (b) √8,281; and (c) √6.084 by long division.

问题7：找到（a）√144的平方根； （b）√8,281的平方根； （c）√6,084的平方根。

Answer # 7: Answers to these three square root questions can be viewed from the three videos below:

答案7：可以从以下三个视频中查看这三个平方根问题的答案：

(a) 144. https://youtu.be/bq-qKuRG_P4

(b) 8,281. https://youtu.be/Ga1_wuLz0QM

(c) 6,084. https://youtu.be/gkx5krbE2Pw

Question # 8: Solve for the unknown in the following equations x, y, a, b, p:

问题# 8：在以下的方程式中，求解未知数 x, y, a, b, p：

# 8a. xy = 56 and x - y = 10

First, find the factors of 56. These are: 1 and 56; 2 and 28; 4 and 14; 7 and 8.

首先，找到56的因数。分别是：1 和 56； 2 和 28; 4 和 14； 7 和 8。

Second, look at these factors and select the pair with a difference of 10.

其次，查看这些因素并选择相差10的那对。

Yes, that pair is 4 and 14.

是的，那对是 4 和 14。

So the answers are x = 14 and y = 4. x - y = 10 and x multiplied by y = 56.

因此答案是 x = 14 和 y =4。x - y = 10 且 x 乘以 y = 56。

# 8b. Similarly, if we have ab = 30 and a + b = 11, we find the factors in the same way.

＃8b。类似地，如果我们有 ab = 30 和 a + b = 11，我们将以相同的方式找到因子。

1 and 30; 2 and 15, 3 and 10; 5 and 6. Then a = b = either 5 or 6. Then a multiplied by b = 30 and a + b = 11.

1 和 30; 2 和 15、3 和 10； 5 和 6。然后 a = b = 5 或 6。然后 a 乘以 b = 30 且 a + b = 11。

# 8c. p2 = 625. p = √625. The answer is 25 by the long division method. 2 x 2 = 4; [So take 2 as the first digit of the answer] 6 - 4 = 2; Then double the remainder 2 to get 4; bring down the next pair 25 to make the solution of 225; then 45 x 5 = 225. So the second digit in the answer is 5.

＃8c。 p2 =625。p =√625。用长除法的答案是25。 2 x 2 = 4; [因此，将2作为答案的第一位数字] 6-4 = 2; 然后将余数2加倍，得到4；否则，加 2。调低下一对25以获得225的解;那么45 x 5 =225。因此答案中的第二位数字是5。

Question # 9: If AUS $120.00 = NZ $124.00 and US$100.00 = AUS $137.00, convert US $315.00 to AUS $ and to NZ $?

问题 # 9：如果 $120.00 澳元 = $124.00 新西兰元，$100.00 美元= $137.00 澳元，则应将 $315.00美元 转换为澳元 和 新西兰元？ 答案可以从下面的方程式中看出。

Break US $315.00. into three smaller sums of $300, $10, $5.

US $100.00 = AUS $137.00. US $300.00 = 3 x AUS $137.00 = AUS $311.00

Plus: US $10.00 = AUS $13.70. (one-tenth of AUS $137.00)

Plus: US $ 5.00 = half of US $10.00 = AUS $ 6.85. Total: AUS $ 331.55

Now, we can do the same steps to convert US$315.00 or AUS $331.55 to NZ$.

AUS $120.00. = NZ $124.00.

AUS $100 = NZ (124 divided by 120 and multiplied by 100) = NZ$ 103.33

Break up AUS 331.55 into smaller sums of AUS $300; AUS $30; AUS $1.00; AUS $0.50; and AUS $0.05

AUS $300.00 = 3 x NZ $103.33 = NZ $309.99

AUS $30.00 = 10% of AUS $300 = NZ $31.00

AUS $ 1.00 = NZ $1.03

AUS $ 0.50 = half of AUS $1.00 = NZ $ 0.51

AUS $ 0.05 = NZ $ 0.05. Total: NZ $ 342.58 for US$ 315.00 or AUS $331.55

Question # 10: Find the area in square centimeters of the pink-colored patch in the attached photograph.

问题 # 10：在所附照片中, 找粉红色补丁的平方厘米面积。

Answer # 10: 答案 # 10：

Area of Entire square: 34 cm multiplied by 34 cm = 1,156 square centimetres.

Deduct three Cut-offs. First, the right-angled triangle on the left hand top corner. Area = half base x height. Area = half (0.5) x 20 x 22 = 220 square centimetres.

Deduct the quadrant at the left-hand bottom corner. This is one-quarter of Pi x radius squared. Area = 0.25 x 3.147 x 8 x 8 = 50.352 square centimetres.

Deduct the quadrant at the right-hand top corner. Area = 0.25 x 3.147 x 10 x 10 = 78.675 square centimetres.

Total deduction = 220 + 50.352 + 78.675 = 349.027square centimetres.

Answer therefore is 1,156 - 349.027 = 806.973 square centimetres.