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Question

At the start of the year, Jonathan and Sarah received daily allowance from their mother. Jonathan received a fixed amount of allowance each day. Sarah received 30 cents more than Jonathan for daily allowance each day. Jonathan saved 90 cents more than Sarah each day. A number of days later, Sarah spent $120 and Jonathan spent $48.

a) How many days did Sarah take to spend $120?

b) How much was Sarah spending each day, given that she spent the same amount of money each day?

 

TIA!

Answer

Since Sarah has $0.30 more allowance than Jon each day, Sarah has to spent $1.20 more than Jon each day for Jon to save $0.90 more.

Difference in total spending = 120 – 48 = $72
A ) Total days needed = 72/1.2 = 60 days ###

B) each day she spent = $120 / 60 = $2 ###

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post-19991-1

Step.  1: draw the green portion which shows the total allowance. Sarah should have an additional 30.

Step 2:add the blue savings in.  Jon should have additional 90. ( savings shod be added in as its part of the allowance.

step 3: the rest of the empty boxes in the model would be what’s spent. Denoted by sp.

step 4: isolate the spending part. ( drawn by yellow lines ).it  clearly shows Sarah spends $1.20 more.

Hence,

difference :$120-$48 = $72.

$72 / $1.20 = 60 days (a)

(b) $120 / 60 = 2 

 

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Thanks for the replies.

Enders’ answer is the simplest, thus easiest for P5 kid to understand.

But still cannot understand the part on 30 cents +90 cents=$1.20… Sarah “spent” or “saved” that? Or the same?

Algebra is still pretty confusing at this stage.

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post-19976

Hope this model for a day’s allowance and spending will help to explain why Sarah has to spent $1.20 more than Jon, such that Jon will saved $0.90 more.

Hope your DC will be able to visualize it in his/her head with more practices.

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Hi Daddy D, I agree with you that Ender’s workings is the most straight-forward. The confusion is due to too much information (allowances, savings, spending, more, less, etc). If your message is a question here, I’ll try to answer below.

We can think simple first: Suppose each of them receives the same amount of daily allowance, so if Thrifty Jonathan saves $0.90 more, on the flip side, Spendthrift Sarah spends $0.90 more.

However, Sarah gets additional $0.30 more from their mother. So the only reason Jonathan can still save $0.90 more would mean Sarah spending away this additional $0.30. So Sarah spends $1.20 more than Jonathan on a daily basis.

 

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Quoting Khong Pek Mao’s answer below:

This is a Matrix question. 

  Save        Spent      Total      
    Sarah    1u 120  1u + 120  
Jonathan   1.9u 48  1.9u + 48  

1u + 120 = 1.9u + 48 + 0.3u

120 – 48 = 2.2u – 1u 

1.2u = 72

1u = 60 days 

b)  120 / 60 = $2 

Unquote

Hi Khong, how do you get “1.9u”? If it is because Jonathan saves 90cent more, this info is not equivalent to 1.9u savings by Jonathan, where Sarah saves “1u” and “u” is the no. of days.

Let’s say Sarah saves $x per day, the summary table should be as below:

  Save        Spent      Total allowances  
    Sarah    ($x) u days 120  xu + 120  
Jonathan  ($x+0.9) u days   48  xu + 0.9u + 48  

xu + 120 = xu + 0.9u + 48 + 0.3u

[where 0.3u is the extra allowances to “add to J” so the allowances equate)

120 – 48 = 1.2u [where xu are cancelled out]

1.2u = 72

1u = 60 days 

b)  120 / 60 = $2 

So in this sense, Sarah’s daily savings ($x) or Jonathan’s daily savings ($x+$0.90) is not important in the solution (question do not have the information to solve this $x).

Key deduction from the problem is that Jonathan is spending $1.20 less than Sarah per day… because he saves $0.90 more (=spend $0.90 less) but also need to “save up” the $0.30 difference (=spend $0.30 less) as their “unfair” mother is giving $0.30 more to Sarah each day. And then, applying this daily difference to the total difference in their accumulated spendings like Ender’s workings.

2 Replies 1 Like

Hi, 

Thank you for your comment.

When we use 1u for Sarah, this is the total amount she save in a certain day.

So Jonathan would have save 1u + 0.9u = 1.9u, since he save $0.9 more each day.

In some problem sum, like both person give away equal amount of money, that equal amount is ignored. That is simply when we draw modal, we only concentrate on the more than end instead of the equal end. In this case, whatever the 1u will be, it is contra off from each side of the equation.

As in Primary school level, we shall not use more than 1 unknown (like x or p ) as their knowledge of algebra is not that good. The school only teach the substitution of number into an expression.

I have seen a lot of solving methods involving Simultaneous Equation by writing out like 1a + 2m = $5. Other than those experts in problem solving pupils, I do not think that those below average pupils can understand this equation.

 

1 Reply 1 Like

Hi Khong, thanks for feedback and I do understand what you mean. However, even for algebra beginners, the denotation of unknown needs to be mathematically accurate at all times.

I would like to suggest the below because to me, mathematically, “1u” is *not equivalent to [1 part of Savings], when 0.9u denotes the extra savings Jonathan saves in “u” no. of days:

  Save        Spent      Total allowances  
    Sarah    1 part of Savings (in u days) 120  1 part of Savings + 120  
Jonathan 

1 part of Savings (in u days)

+ 0.9u (extra in u days)  

48  1 part of Savings + 0.9u + 48  

[1 part of Savings] + 120 = [1 part of Savings] + 0.9u + 48 + 0.3u

*if u=10days, it is correct to say Sarah has 1 part of Savings and Jonathan has 1 part of Savings + $9. But because of wrong algebra, one may be misled to think that Sarah saves $10 (1u) and Jonathan saves $19 (1.9u) in 10 days.

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OK.  Thank.

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post-19907

Maybe that is why algebra is so tricky and not encouraged at Primary level when it gets more complex. Might be clearler in model here.

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Just for interest only – to understand through algebra. Below is summary table of what I mean the key deduction of daily difference in spending.

 

Daily Allowances

(A)

Daily Savings

(B)

Daily Spending

(A)-(B)

 Accumulated Spending
 Sarah    $p+$0.30 $x  $(p-x) +$0.30 $120
Jonathan  $p $x+$0.90  $(p-x) – $0.90 $48
Difference     $1.20 $72
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This is a Matrix question. 

  Save        Spent      Total      
    Sarah    1u 120  1u + 120  
Jonathan   1.9u 48  1.9u + 48  

1u + 120 = 1.9u + 48 + 0.3u

120 – 48 = 2.2u – 1u 

1.2u = 72

1u = 60 days 

b)  120 / 60 = $2 

 

 

 

 

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