Problem with simulataneous equations :( (1 Viewer)

[AnandDNA]

Im stuck on some easy simulataneous worded problems. These are the 3 im having trouble with:

1. At a meeting of the members of a certain club, a proposal was voted on. If 357 members voted and the proposal was passed by a majority of 21, how many voted for and how many voted against the proposal?

2. Two people are 16km apart on a straight road. They start walking at the same time. If they walk towards each other, then they will meet in 2hrs., but if they walk in the same direction (so that the distance between them is decreasing), then they will meet in 8hrs. find their walking speeds?

3. A certain integer is between 10 and 100. its value is 8 times the sum of it digits and if it is reduced by 45, its digits are reversed. find the interger.?

So if anyone can help it will be much appreciated

 

[Mark576]

3. Let the number be represented by a+10b;
a+10b=8(a+b) (1)
a+10b-45=b+10a (2)

From (1): 7a = 2b therefore a = 2/7 b

From (2): 9a = 9b - 45
a = b - 5
b = a + 5
b = 2/7 b + 5
therefore 5/7 b = 5 => b = 7 and a = 2

Checking this: 72 is between 10 and 100, sum of its digits is 9, multiplied by 8 gives it's value and 72 - 45 = 27. Hopefully that helps you with it!

 

[ssglain]

Most of these questions simply require you to translate chunks English text into maths. This is usually the part that students find difficult some times. Once you have obtained the mathematical description of the given conditions, some simultaneous algebraic crunching will get you the answers.

1. Let the number of people who voted for the proposal be x and the number who voted against be y.

"357 members voted" means that there were a total of 357 people at this meeting some of who voted for and other voted against. Translated into maths this means:
x + y = 357 ...(1)

"the proposal was passed by a majority of 21" means that the number of people who voted for the proposal was 21 more than the number who voted against it. Translated into maths this means:
x = y + 21
i.e. x - y = 21 ...(2)

(1)+(2): 2x = 378 --> x = 169

Put x = 169 in (2): 169 - y = 21 --> y = 148

.: 169 people voted for the proposal and 148 voted against it.

 

[ssglain]

2. Let the walking speed of person A be a km/h and person B be b km/h.

"If they walk towards each other, then they will meet in 2hrs" means after person A has walked for 2 hrs at a km/h and person B for 2 hrs at b km/h towards each other, the distance between them is 0. So together them they have covered 16km of distance.
2a + 2b = 16
i.e. a + b = 8 ...(1)

"if they walk in the same direction ... then they will meet in 8hrs" is a lot more tricky. Think about it carefully though - suppose person A walked towards B and person B just kept walking away and after 8 hours person A eventually catches up. This means that at a km/h, person A was able to walk the 16 km that was originally between them as well as whatever amount of distance that person B walked in those 8 hours.
8a = 8b + 16
i.e. a - b = 2 ...(2)

(1)+(2): 2a = 10 --> a = 5
Put a = 5 in (2): 5 - b = 2 --> b = 3

.: Person A walks at 5 km/h and person B walks at 3 km/h.

 

[AnandDNA]

Thanks for the help Mark and ssglain

But referring to the first question about the club proposal the answer is not right because 148+169 doesnt =357.

But i still worked it out by using the elimination method so
the last line should be 2y=336 --> y=168

sub in y value into (1): x+168=357 --> x=189

Test 189+168 =357
189-168 =21

and sorry but i dont understand how you worked out to represent the number with a+10b
But thanks for the help on the questions ssglain and mark

 

[ssglain]

My mental arithmetic is failing me. 378/2 = 189 not 169. LOL. Can't live without having a calculator these days.

Mark let b represent the digit in tens and a represent the digit in ones, so the magnitude of this two-digit number becomes a +10b.

 

[kaz1]

At a meeting of the members of a certain club, a proposal was voted on. If 357 members voted and the proposal was passed by a majority of 21, how many voted for and how many voted against the proposal?

x= voted against , x +21 = voted for

so 2x + 21 = 357
2x = 336
x= 168 ( voted against)
x + 21 = 189 ( voted for)



this method was easier for me

 

[ssglain]

At a meeting of the members of a certain club, a proposal was voted on. If 357 members voted and the proposal was passed by a majority of 21, how many voted for and how many voted against the proposal?

x= voted against , x +21 = voted for

so 2x + 21 = 357
2x = 336
x= 168 ( voted against)
x + 21 = 189 ( voted for)



this method was easier for me

Yes, I personally prefer that method. However, if a question specifies the use of simultaneous equations then do as they ask.

 

[Slidey]

At a meeting of the members of a certain club, a proposal was voted on. If 357 members voted and the proposal was passed by a majority of 21, how many voted for and how many voted against the proposal?

x= voted against , x +21 = voted for

so 2x + 21 = 357
2x = 336
x= 168 ( voted against)
x + 21 = 189 ( voted for)



this method was easier for me

That method is implicitly using simultaneous equations anyway.

Let y mean 'voted for' and x mean 'voted against', then: y=x+21, and x+y=357.