Both Party A and Party B are driving at a constant speed on the circular road. If they start from the same place and go in the opposite direction at the same time, then they will meet every 2 minutes. If you go in the same direction, meet every 6 minutes. It is known that A runs faster than B. How many laps do A and B run per minute?

Analysis: the opposite direction is a problem of meeting, and the same direction is a problem of catching up.

Solution: Method 1, (understandable)

A speed +B speed = 1/2

A speed -B speed = 1/6

That's it: a speed =( 1/2+ 1/6)÷2.

=2/3÷2

= 1/3

B speed = 1/2- 1/3.

= 1/6

A: A runs 1/3 laps per minute, and B runs 1/6 laps per minute.

Binary linear equation:

Law two. Let the speed of A be x cycles per minute and the speed of B be y cycles per minute to obtain the equation:

1/(x+y)=2

1/(x-y)=6

Simplify:

x+y= 1/2

x-y= 1/6

It is easy to get x= 1/3 and y= 1/6, so A runs 1/3 laps per minute and B runs 1/6 laps per minute.

Teacher Wang got off work at 6 pm and went to the supermarket to buy food. At this time, the angle between the hour hand and the minute hand on the clock is 1 10. When he got home at 7 o'clock, he found that the angle between the hour hand and the minute hand on the clock was still 1 10. Can you work out how long it took Mr. Wang to shop?

Analysis:

Arithmetic method:

The minute hand turns 6 degrees per minute and the hour hand turns 0.5 degrees per minute.

According to the catch-up problem,

The minute hand fell behind 1 10 degrees, and finally exceeded 1 10 degrees.

Need to catch up to 220 degrees.

Catch up in minutes (6-0.5)=5.5 degrees.

(110+110) ÷ (6-0.5) = 40 minutes.

Mr. Wang spent 40 minutes shopping.

Equation:

Solution: Suppose it takes X minutes for Mr. Wang to shop.

(6-0.5)x = 1 10+ 1 10

x=40

Mr. Wang spent 40 minutes shopping.

In the regular pentagon ABCDE, m and n are points on DE and EA respectively, and BM and CN intersect at point O. If ∠ bon = 108, does the conclusion BM=CN hold? If yes, please give proof; If not, please explain why.

When ∠ bon = 108. BM=CN also holds.

Prove; Connect BD and CE as shown in fig. 5.

In △BCI) and △CDE.

BC = CD，∠BCD=∠CDE= 108，CD=DE

∴δbcd≌δCDE

∴BD=CE，BDC=∠CED，DBC=∠CEN

∠∠CDE =∠dec = 108，∴∠BDM=∠CEN

∠∠OBC+∠ECD = 108，∠OCB+∠OCD= 108

∴∠MBC=∠NCD

∠∠DBC =∠ECD = 36，∴∠ DBM =∠ ECN。

∴δbdm≌δCNE ∴bm=cn

Equation arithmetic is also a problem, and the exam is scored. The reference is all geometry questions, take your time ~ ~