Ratio question: If there is 8kg of water, how much concrete will the mix make?

[Bananaman]

Apologies if this is in the wrong section, I wasn't sure where to put it. A cement mixture calls for cement, sand and water in the ratio of 1:2:4.


If there is 8kg of water, how much concrete will the mix make? Based on that I can work out how much of the other ingredients there are, 2kg cement, 4kg sand and 8kg water. How do I figure out how much concrete the mix can make tho? If they said ''if this amount of cement, sand and water makes this much concrete, how much concrete can you make if you have 8kg water'', then I could work it, but how can I do it without any reference point, i am confused...

 

[Dr.Peterson]

Apologies if this is in the wrong section, I wasn't sure where to put it. A cement mixture calls for cement, sand and water in the ratio of 1:2:4.


If there is 8kg of water, how much concrete will the mix make? Based on that I can work out how much of the other ingredients there are, 2kg cement, 4kg sand and 8kg water. How do I figure out how much concrete the mix can make tho? If they said ''if this amount of cement, sand and water makes this much concrete, how much concrete can you make if you have 8kg water'', then I could work it, but how can I do it without any reference point, i am confused...

In a sense, this is not a math question, but perhaps a physics or chemistry question!

You have solved the math: you have to mix 2 kg cement, 4 kg sand, and 8 kg water. When they are mixed, will the mass change? No; this is called the conservation of mass (or matter). So the total mass of the concrete is 2+4+8 = 14 kg. (This assumes that the given ratio is, as you assume, according to mass.)

Now, if the question had been about a ratio of volumes rather than mass, you would have to determine whether the mixture retains the same total volume. Volume is not always conserved; salt added to water will result in little more volume than that of the water, because the salt molecules/atoms "hide" among those of water. But I think you know that sand will not dissolve in water; I think the same is true of cement. But that is not an issue in the problem as it stands.

 

[Bananaman]

In a sense, this is not a math question, but perhaps a physics or chemistry question!

You have solved the math: you have to mix 2 kg cement, 4 kg sand, and 8 kg water. When they are mixed, will the mass change? No; this is called the conservation of mass (or matter). So the total mass of the concrete is 2+4+8 = 14 kg. (This assumes that the given ratio is, as you assume, according to mass.)

Now, if the question had been about a ratio of volumes rather than mass, you would have to determine whether the mixture retains the same total volume. Volume is not always conserved; salt added to water will result in little more volume than that of the water, because the salt molecules/atoms "hide" among those of water. But I think you know that sand will not dissolve in water; I think the same is true of cement. But that is not an issue in the problem as it stands.

Thanks for the reply, that's very helpful. The method used by the maths book is 8 x 7 ÷ 4 = 14kg, it looks like they have multiplied the 8kg water by the total number of parts in the ratio and then divided that by the ratio of water. A much more convoluted method then 2 + 4 + 8...

 

[Dr.Peterson]

Thanks for the reply, that's very helpful. The method used by the maths book is 8 x 7 ÷ 4 = 14kg, it looks like they have multiplied the 8kg water by the total number of parts in the ratio and then divided that by the ratio of water. A much more convoluted method then 2 + 4 + 8...

What you did was more work, but is perhaps more straightforward -- both are perfectly good.

The traditional way to handle this sort of ratio is what the book did: given the ratio 1:2:4, the total is 1+2+4 = 7 "parts", so the ratio of concrete to water is 7:4. Then you can just solve 7:4 = x:8 as usual. That's what they did, without having to separately find the amount of each component.

 

[Bananaman]

What you did was more work, but is perhaps more straightforward -- both are perfectly good.

The traditional way to handle this sort of ratio is what the book did: given the ratio 1:2:4, the total is 1+2+4 = 7 "parts", so the ratio of concrete to water is 7:4. Then you can just solve 7:4 = x:8 as usual. That's what they did, without having to separately find the amount of each component.

Apologies for my difficulty understanding, you lost me at the 7:4 part, why are you putting the total number of parts next to the ratio of water?

 

[Dr.Peterson]

Apologies for my difficulty understanding, you lost me at the 7:4 part, why are you putting the total number of parts next to the ratio of water?

Because you want to find the total mass (that is, the mass of concrete).

You have cement:sand:water = 1:2:4. So you need 1 "part" (whatever quantity that is) of cement, 2 of sand, and 4 of water. That's a total of 7 "parts"; for every 4 parts of water, you have 7 parts of concrete (the mixture). So the ratio of concrete to water is 7:4.

 

[Bananaman]

Because you want to find the total mass (that is, the mass of concrete).

You have cement:sand:water = 1:2:4. So you need 1 "part" (whatever quantity that is) of cement, 2 of sand, and 4 of water. That's a total of 7 "parts"; for every 4 parts of water, you have 7 parts of concrete (the mixture). So the ratio of concrete to water is 7:4.

Ok I understand the ratio 7:4 now thank you. So 7:4 = 14:8, that's very easy to understand. The book only tells me to do 8 x 7 ÷ 4 without any further explanation, which appears to me like a different method from the one your suggesting. The formula suggested in the book (mass of water x total parts ÷ ratio of water) obviously works, I am just struggling to understand why it works, but I guess that's like asking why does 2+2 = 4. I'm probably just overthinking it, I can always stick with the method I am comfortable with.

 

[Dr.Peterson]

Ok I understand the ratio 7:4 now thank you. So 7:4 = 14:8, that's very easy to understand. The book only tells me to do 8 x 7 ÷ 4 without any further explanation, which appears to me like a different method from the one your suggesting. The formula suggested in the book (mass of water x total parts ÷ ratio of water) obviously works, I am just struggling to understand why it works, but I guess that's like asking why does 2+2 = 4. I'm probably just overthinking it, I can always stick with the method I am comfortable with.

They are doing exactly what I did, adding up the total parts and using that, but as a formula rather than explicitly as a new ratio. They just didn't say enough to make it clear to someone who can't read their minds. (I think you're saying they did state that formula, but you didn't quote it before)

Your way was fine; what I said actually may be something I have never said explicitly as I did. I don't think of it as a formula as they wrote it. In part, I think in terms of a picture:

[FONT=courier new] 1  2     4
+-+---+-------+
|C| S |   W   |
+-+---+-------+
       7[/FONT]

so that the water is 4/7 of the whole. The whole is therefore 7/4 of the 8 kg, or 14 kg. For me, that all comes to mind without having to say it. For you, whatever comes to mind is better than something that doesn't!

 

[Bananaman]

They are doing exactly what I did, adding up the total parts and using that, but as a formula rather than explicitly as a new ratio. They just didn't say enough to make it clear to someone who can't read their minds. (I think you're saying they did state that formula, but you didn't quote it before)

Your way was fine; what I said actually may be something I have never said explicitly as I did. I don't think of it as a formula as they wrote it. In part, I think in terms of a picture:

[FONT=courier new] 1  2     4
+-+---+-------+
|C| S |   W   |
+-+---+-------+
       7[/FONT]

so that the water is 4/7 of the whole. The whole is therefore 7/4 of the 8 kg, or 14 kg. For me, that all comes to mind without having to say it. For you, whatever comes to mind is better than something that doesn't!

They didn't state the formula, I deduced it myself from the 8 x 7 ÷ 4 = 14kg, which they gave as the solution. I understand your explanation perfectly, its just the formula in the book that I struggled with. I sense some frustration in your last post so I'll leave it there, I don't want to overstay my welcome.

 

[mmm4444bot]

͏͏
You're welcome to stay as long as you like. :cool:

… The book only tells me to do 8 x 7 ÷ 4 without any further explanation …

… I deduced … The formula suggested in the book (mass of water x total parts ÷ ratio of water) …

I think you meant "suggested by the book" because a formula is not actually in the book (only an expression).

For the author's formula, you guessed: concrete mass = (mass of water x total parts ÷ ratio of water). Yet, to somebody else, the book's expression 8×7÷4 might suggest the following formula -- because, in the book's example, this formula also "works". :wink:

concrete mass = (mass of water × total parts) ÷ (mass of water to sand parts)

It's easy to misinterpret expressions, when trying to reverse-engineer a conclusion. If there was no prior instruction about proportions or solving them, then it's too bad the author didn't explain or provide the formula, in their "maths book".

---

You've written "ratio of water" to represent 4, but that phrase doesn't provide the two values used in the division to get 4.

A ratio always compares two quantities, like "ratio of water to sand", for example.

We have three notations for expressing a ratio:

water to sand

water : sand

water/sand

---

Check this out:

concrete mass = cement mass ÷ (cement parts/total parts)

concrete mass = water mass ÷ (water parts/total parts)

concrete mass = sand mass ÷ (sand parts/total parts)

---

Note: Dividing by a ratio is the same as multiplying by its reciprocal. To write a fraction's reciprocal, we "flip it". For example, the reciprocal of 4/7 is 7/4.

Therefore, 8 ÷ (4/7) is the same as 8 × (7/4)

8 × (7/4) can also be expressed as 8 × 7 ÷ 4

As to its origin, therefore, the expression 8 × 7 ÷ 4 is ambiguous.

---

When two ratios are equal, we call that equation a proportion:

4/7 = 8/14

The formulas above come from solving the following proportions to get expressions for the total mass (i.e., kg of concrete):

(cement parts/total parts) = (cement mass/total mass)

(water parts/total parts) = (water mass/total mass)

(sand parts/total parts) = (sand mass/total mass)

Cheers :cool:

 

[Bananaman]

͏͏
You're welcome to stay as long as you like. :cool:

I think you meant "suggested by the book" because a formula is not actually in the book (only an expression).

For the author's formula, you guessed: concrete mass = (mass of water x total parts ÷ ratio of water). Yet, to somebody else, the book's expression 8×7÷4 might suggest the following formula -- because, in the book's example, this formula also "works". :wink:

concrete mass = (mass of water × total parts) ÷ (mass of water to sand parts)

It's easy to misinterpret expressions, when trying to reverse-engineer a conclusion. If there was no prior instruction about proportions or solving them, then it's too bad the author didn't explain or provide the formula, in their "maths book".

Yes that's what i meant, i misspoke, sorry.

͏͏

---

You've written "ratio of water" to represent 4, but that phrase doesn't provide the two values used in the division to get 4.

A ratio always compares two quantities, like "ratio of water to sand", for example.

We have three notations for expressing a ratio:

water to sand

water : sand

water/sand

Yes, maybe I should of said something like x parts water instead of ratio of water when describing the formula.

---

͏͏

Check this out:

concrete mass = cement mass ÷ (cement parts/total parts)

concrete mass = water mass ÷ (water parts/total parts)

concrete mass = sand mass ÷ (sand parts/total parts)

---

Note: Dividing by a ratio is the same as multiplying by its reciprocal. To write a fraction's reciprocal, we "flip it". For example, the reciprocal of 4/7 is 7/4.

Therefore, 8 ÷ (4/7) is the same as 8 × (7/4)

8 × (7/4) can also be expressed as 8 × 7 ÷ 4

As to its origin, therefore, the expression 8 × 7 ÷ 4 is ambiguous.

---

When two ratios are equal, we call that equation a proportion:

4/7 = 8/14

The formulas above come from solving the following proportions to get expressions for the total mass (i.e., kg of concrete):

(cement parts/total parts) = (cement mass/total mass)

(water parts/total parts) = (water mass/total mass)

(sand parts/total parts) = (sand mass/total mass)

Cheers :cool:

I understand now, thank you. Am I correct in assuming that the only way to do 8 ÷ (4/7) is to first do 4 ÷ 7, then 8 divide by the answer? If that is the case then 8 x 7 ÷ 4 seems like the simpler method.

 

[mmm4444bot]

Am I correct in assuming that the only way to do 8 ÷ (4/7) is to first do 4 ÷ 7, then 8 divide by the answer?

The answer depends on whether you're using paper 'n pencil, a basic calculator, or something fancier (eg: scientific calculator, math software).

If you're using a basic calculator, the answer is yes. You will get the wrong answer, if you push:

8 ÷ 4 ÷ 7 =

If you're using a decent scientific calculator, then grouping symbols are available, and you may push:

8 ÷ ( 4 ÷ 7 ) =

If you're using paper 'n pencil, it's very obnoxious to do long division with a fraction. Switch from division to multiplication (by the reciprocal):

8 × 7/4 =

8/4 × 7 =

2 × 7 =

14

Notice how I divided 8 by 4, first (instead of multiplying by 7, first)? I did that because 8/4 is easier to do in my head than 56/4.

When a simple expression contains no grouping symbols, and the operations are multiplication and division only, it's safe to change the order of operations. (See the commutative and associative properties of multiplication.)

The following four expressions are equal, given three numbers A,B,C.

A × B ÷ C

B × A ÷ C

A ÷ C × B

B ÷ C × A

 

[Bananaman]

The answer depends on whether you're using paper 'n pencil, a basic calculator, or something fancier (eg: scientific calculator, math software).

If you're using a basic calculator, the answer is yes. You will get the wrong answer, if you push:

8 ÷ 4 ÷ 7 =

If you're using a decent scientific calculator, then grouping symbols are available, and you may push:

8 ÷ ( 4 ÷ 7 ) =

If you're using paper 'n pencil, it's very obnoxious to do long division with a fraction. Switch from division to multiplication (by the reciprocal):

8 × 7/4 =

8/4 × 7 =

2 × 7 =

14

Notice how I divided 8 by 4, first (instead of multiplying by 7, first)? I did that because 8/4 is easier to do in my head than 56/4.

When a simple expression contains no grouping symbols, and the operations are multiplication and division only, it's safe to change the order of operations. (See the commutative and associative properties of multiplication.)

The following four expressions are equal, given three numbers A,B,C.

A × B ÷ C

B × A ÷ C

A ÷ C × B

B ÷ C × A

Sorry for the delayed reply, my internet has been down for a few days. I understand now, thanks for the help.