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smee
07-23-2009, 07:44 AM
Home work for my kid - I put 4 pencils into each jar and have 1 jar left over. I put 3 pencils into each jar and have 1 pencil left over. How many jars and how many pencils ?
Need help urgently, Im quite retarded with things like this !!!

Subhotosh Khan
07-23-2009, 08:45 AM
Home work for my kid - I put 4 pencils into each jar and have 1 jar left over. I put 3 pencils into each jar and have 1 pencil left over. How many jars and how many pencils ?
Need help urgently, Im quite retarded with things like this !!!

There are different ways do this problem - what grade level is your kid? Has s/he done Algebra - solution of linear equations?

soroban
07-23-2009, 09:20 PM
\text{Hello, smee!}

\text{Here's one approach . . .}


I put 4 pencils into each jar and have 1 jar left over.
I put 3 pencils into each jar and have 1 pencil left over.
How many jars and how many pencils?
\text{Let: }\:\begin{array}{ccc}J &=& \text{number of jars} \\ P &=& \text{number of penciles} \end{array}


\text{4 pencils per jar, 1 jar left over: }\quad\begin{array}{ccccccc} /\!\!/\!\!/\!\!/ & /\!\!/\!\!/\!\!/ & /\!\!/\!\1\!/\!\!/ && /\!\! /\!\! /\!\! / \\ \bigsqcup & \bigsqcup & \bigsqcup & \hdots & \bigsqcup && \bigsqcup
. . . . . . . . . . . . . . . . . . . . . . . . \underbrace{\qquad\qquad\qquad\qqiuad\qquad\quad\; }_{J-1\text{ jars}}
. . \text{We have: }\;P \;=\;4(J-1)\quad\bf{[1]}


\text{3 pencils per jar, 1 pencil left over: }\quad \begin{array}{ccccccc}/\!\! /\!\! / & /\!\! /\!\! / & /\!\! /\!\! / && / \!\! / \!\! / & / \\ \bigsqcup & \bigsqcup & \bigsqcup & \hdots & \bigsqcup\\ \end{array}
. . . . . . . . . . . . . . . . . . . . . . . . . .\underbrace{\qquad\qquad\qquad\qquad\quad\;}_{J\t ext{ jars}}
. . \text{We have: }\;P \:=\:3J + 1\quad\nf{[2]}


\text{Equate }\bf{[1]}\text{ and }\bf{[2]} \!: . 4(J-1) \:=\:3J + 1 \quad\Rightarrow\quad 4J - 4 \:=\:3J + 1 \quad\Rightarrow\quad J \:=\:5

\text{Substitute into }\nf{[2]}\!: . P \:=\:3(5)+1 \:=\:16


\text{Theefore, there are 5 jars and 16 pencils.}

Aladdin
07-24-2009, 08:44 AM
j = \frac{p}{4} + 1

p = 3j + 1

Solve the system of equations.

Subhotosh Khan
07-24-2009, 08:46 AM
Another way,

There are four pencils in each jar.

Then the number of pencils could be = 4, 8, 12, 16, 20, 24, 28.....

(There are atleast 2 jars - since 1 jar was left empty above)

There are three pencils in each Jar + 1. (There are atleast 2 jars - since 1 jar was left empty)

Then the number of pencils could be = 7, 10, 13, 16, 19, 22, 25.....

For a given number of jars - the only match is 5 jars and 16 pencils.

Denis
07-24-2009, 10:08 AM
pencils per jar = a ; leftover jars = u
pencils per jar = b ; leftover pencils = v

JARS = (au + v) / (a - b)

I put 13 pencils into each jar and have 3 jars left over. [a = 13, u = 3]
I put 9 pencils into each jar and have 5 pencils left over. [b = 9, v = 5]
How many jars have I got?

Try that, Mommy Smee :wink: