Simon's copy of Q5 (Calculating wages using algebraic equations)

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Simon Thomas 6 years, 1 month ago

Created this as a copy of Q5 (Calculating wages using algebraic equations).

Name	Status	Author	Last Modified
Q5 (Calculating wages using algebraic equations)	Ready to use	TEAME CIT	06/10/2016 16:14
Simon's copy of Q5 (Calculating wages using algebraic equations)	draft	Simon Thomas	25/03/2019 09:52

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Question statement

Calculate the following, to the nearest cent!

Give any introductory information the student needs.

			Name	Type	Generated Value

num11	integer	9
num12	integer	7
wages1	integer	8954
num13	integer	75
ans11	rational	9479/16
ans12	rational	8279/16
ans21	integer	7
ans22	integer	5
num21	integer	12
num22	integer	520
num23	integer	770
wages2	integer	7490
num31	integer	6
num32	integer	11
wages3	integer	7275
num33	integer	76
ans31	rational	8111/17
ans32	rational	6819/17
num41	integer	5
num42	integer	7
wages41	integer	9402
num43	integer	4
wages42	integer	2415
ans41	rational	2673/13
ans42	rational	20703/13

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$\var{num41}$ tradesmen and $\var{num42}$ labourers earn $€\var{wages41}$ per week while 1 tradesman and $\var{num43}$ labourers earn $€\var{wages42}$ per week. Find the earnings for 1 tradesman and 1 labourer.

Tradesmen: €Gap 0.

Labourer: €Gap 1.

Advice

(a)

Let $L$ be the wage for 1 labourer, and let $T$ be the wage for 1 tradesman

Then we can set up two simultaneous equations as follows:

$\var{num41}T+\var{num42}L=\var{wages41}$ (Equation 1)

$T + \var{num43}L=\var{wages42}$ (Equation 2)

Multiplying equation 2 by $\var{num41}$ gives:

$\var{num41}T+\var{num41*num43}L=\var{num41*wages42}$ (Equation 3)

Subtracting equation 1 from equation 3 eliminates $T$ and gives:

$\var{num41*num43}L-\var{num42}L=\var{num41*wages42}-\var{wages41}$

which we can solve for $L$ as follows:

$\var{-num42+num41*num43}L=\var{-wages41+num41*wages42}$

$L=\var{precround((wages41-num41*wages42)/(num42-num41*num43),2)}$

We can now find $T$ by substituting this value of $L$ into any of Equations 1-3.

e.g. Substituting into Equation 2 gives:

$T + \var{num43}\times \var{precround((wages41-num41*wages42)/(num42-num41*num43),2)} = \var{wages42}$

$T= \var{precround(wages42-num43*(wages41-num41*wages42)/(num42-num41*num43),2)}$

(b)

$\var{num11}$ tradesmen and $\var{num12}$ labourers earn $€\var{wages1}$ between them to do a job. If a tradesman earns $€\var{num13}$ more than a labourer, calculate the earnings for a tradesman and a labourer.

We can set up two simultaneous equations as follows:

$\var{num11}T+\var{num12}L=\var{wages1}$ (Equation 1)

$T=L+\var{num13}$ (Equation 2)

We can replace $T$ in Equation 1 with $L+\var{num13}$ to obtain:

$\var{num11}(L+\var{num13})+\var{num12}L=\var{wages1}$

Hence:

$\var{num11+num12}L+\var{num11*num13}=\var{wages1}$

$\var{num11+num12}L=\var{wages1-num11*num13}$

$L=\var{precround((wages1-num11*num13)/(num11+num12),2)}$

Using Equation 2 we also obtain:

$T=\var{precround((wages1-num11*num13)/(num11+num12),2)}+\var{num13}=\var{precround(num13+((wages1-num11*num13)/(num11+num12)),2)}$

(c)

$\var{num21}$ workmen on a building site earn a total of $€\var{wages2}$ between them per week. Labourers are paid $€\var{num22}$ per week and Tradesmen are paid $€\var{num23}$ per week. How many of each is employed?

We can set up two simultaneous equations as follows:

$\var{num23}T+\var{num22}L = \var{wages2}$ (Equation 1)

$T+L = \var{num21}$ (Equation 2)

Multiplying equation 2 by $\var{num23}$ gives:

$\var{num23}T+\var{num23}L=\var{num21*num23}$ (Equation 3)

Subtracting equation 1 from equation 3 eliminates $T$ and gives:

$\var{num23}L-\var{num22}L=\var{num21*num23}-\var{wages2}$

which we can solve for $L$ as follows:

$\var{num23-num22}L=\var{-wages2+num21*num23}$

$L=\var{precround((-wages2+num21*num23)/(num23-num22),0)}$

Then substituting this value of $L$ into Equation 2 gives:

$T+\var{precround((-wages2+num21*num23)/(num23-num22),0)} = \var{num21}$

$T = \var{num21-precround((-wages2+num21*num23)/(num23-num22),0)}$

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This question is used in the following exam:

Simultaneous equations by Simon Thomas in Algebra.

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Simon's copy of Q5 (Calculating wages using algebraic equations)

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Taxonomy: mathcentre

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Simon Thomas 6 years, 1 month ago

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Generated value: `integer`

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Simon's copy of Q5 (Calculating wages using algebraic equations)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years, 1 month ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`